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\hyphenation{her-ud-over ek-si-stens-va-ri-ab-le an-dre dob-belt-im-pli-ka-ti-on ob-jekt-kvan-tor de-fi-ni-tions-lem-ma ens-be-tyd-en-de
und-er-af-snit slut-ning-en inde-hol-der for-klar-ing-en si-de-be-ting-el-sen
des-uden be-vis-check-er-en}
\title {\Huge{\AE{}kvivalensrelationer i Logiweb}}
\author {Frederik Eriksen \texttt{(eriksen@diku.dk)}}
\date{22.\ juni 2006}
\maketitle
\newpage
\tableofcontents
\listoffigures
" [ ragged right ] "
\newpage
\section{Indledning}
Denne rapport indeholder et aksiomsystem for en reduceret version af ZF
m\ae{}ngdel\ae{}re, som jeg kalder for ``" [ math system zf end math ] "''. Ud fra dette aksiomsystem vil jeg vise en r\ae{}kke basale s\ae{}tninger inden for udsagnslogik og m\ae{}ngdel\ae{}re. Hovedresultatet er, at en \ae{}kvivalensrelation p\aa{} en m\ae{}ngde " [ math metavar var big set end metavar end math ] " implicit definerer en partition af " [ math metavar var big set end metavar end math ] "\footnote{Jeg vil referere til denne s\ae{}tning som ``hovedresultatet''. Oprindeligt var det ogs\aa{} et m\aa{}l at bevise det modsatte resultat --- at enhver partition implicit definerer en \ae{}kvivalensrelation --- men det har der ikke v\ae{}ret tid til.}. Alt det formelle arbejde i rapporten er gennemf\o{}rt med henblik p\aa{} at bevise dette resultat.
Rapporten er udarbejdet ved hj\ae{}lp af Logiweb, som er et system til verifikation og publicering af tekster, der indeholder formel matematik.
Logiweb har verificeret rapportens beviser og publiceret rapporten p\aa{} WWW-adressen
\noindent \resizebox{\textwidth}{!}{\url{http://www.diku.dk/~grue/logiweb/20060417/home/eriksen/equivalence-relations}.}
Det matematiske indhold af denne rapport er ikke synderligt avanceret; f.eks.\ fylder materialet om \ae{}kvivalensrelationer knap to sider i l\ae{}rebogen \cite{kn:hrba} (s.\ 29--31). Ideen med rapporten er snarere at afpr\o{}ve Logiweb end at redeg\o{}re for en matematisk teori. Derfor vil jeg ikke g\o{}re alt for meget ud af at forklare de matematiske begreber; fokus vil i h\o{}jere grad ligge p\aa{} tekniske forhold omkring formaliseringen.
Rapporten er struktureret som f\o{}lger: Afsnit \ref{sec:logiweb} er en kort beskrivelse af nogle detaljer ved Logiweb, som er nyttige at kende for en l\ae{}ser af et Logiweb-dokument som dette\footnote{Dette afsnit er en revideret udgave af afsnit 2 i \cite{kn:peano}.}. Afsnit \ref{sec:syntaks} og \ref{sec:aksiomsystem} omhandler syntaksen og aksiomsystemet for " [ math system zf end math ] ", og afsnit \ref{sec:makro} indf\o{}rer de makrodefinitioner, jeg vil g\o{}re brug af. S\aa{} kommer vi til lemmaerne og beviserne; vi skal igennem fem afsnit med hj\ae{}lpes\ae{}tninger (afsnit \ref{sec:udsagn}--\ref{sec:lighedslemmaer}), f\o{}r vi endelig kan bevise hovedresultatet i afsnit \ref{sec:hoved}. Afsnit \ref{sec:kon} slutter af med en konklusion p\aa{} det hele.
\section{Lidt om Logiweb} \label{sec:logiweb}
Som n\ae{}vnt i indledningen er denne rapport skrevet ved hj\ae{}lp af Logiweb. Dette betyder dels,
at rapportens formelle indhold er defineret ud fra nogle andre Logiweb-dokumenter, dels at et par afsnit indeholder nogle s\ae{}rlige definitioner og tabeller, og endelig at et par andre afsnit indeholder en del programkode. I dette afsnit vil jeg kort beskrive disse f\ae{}nomener. For en detaljeret beskrivelse af hele Logiweb systemet vil jeg henvise til \cite{kn:base}; her kan man bla.\ l\ae{}se om den bevischecker, der har verificeret rapportens beviser.
\subsection{Formelle konstruktioner} \label{sec:formelkon}
Det formelle indhold af et Logiweb-dokument er sammensat af en r\ae{}kke formelle konstruktioner. En formel konstruktion kan repr\ae{}sentere alt, hvad der har med formel matematik at g\o{}re: Variable, funktioner, lemmaer, beviser, osv. Der er to kilder til konstruktionerne i et Logiweb-dokument: Dels kan man indf\o{}re sine egne konstruktioner, og dels kan man importere konstruktioner fra andre Logiweb-dokumenter. De importerede konstruktioner i denne rapport har to kilder: Dels
\cite{kn:base}, og dels de 3 .pdf filer \cite{kn:check}, som tilsammen udg\o{}r \'{e}t Logiweb-dokument.
\subsection{S\ae{}rlige definitioner og tabeller} \label{sec:saerlig}
Den formelle del af et Logiweb-dokument skrives i et formateringssprog ved navn ``pyk''. Hver formel konstruktion, man arbejder med i dokumentet, har tilknyttet en s\aa{}kaldt ``pyk definition''. Dette er en angivelse af, hvad man skal skrive, hvis man i et andet Logiweb-dokument \o{}nsker at benytte den p\aa{}g\ae{}ldende konstruktion. Hvis man indf\o{}rer en ny konstruktion i sit Logiweb-dokument, er det et krav, at man g\o{}r den tilsvarende pyk definition tilg\ae{}ngelig i dokumentet. Denne rapports pyk definitioner er vedlagt i bilag \ref{sec:pyk}.
Logiweb genererer det f\ae{}rdige dokument ved hj\ae{}lp af det kendte formateringssprog \LaTeX. Derfor har hver formel konstruktion ogs\aa{} tilknyttet en s\aa{}kaldt ``\TeX{} definition'', som angiver, hvordan konstruktionen skal skrives i \LaTeX. Ligesom pyk definitioner skal ogs\aa{} \TeX{} definitioner v\ae{}re tilg\ae{}ngelige i dokumentet. Denne rapports \TeX{} definitioner er vedlagt i bilag \ref{sec:tex}.
Endelig indeholder bilag \ref{sec:prio} en tabel over alle de formelle konstruktioner i denne rapport --- b\aa{}de de importerede og dem, jeg selv har defineret. Denne tabel er f\o{}rst og fremmest med, for at andre Logiweb-dokumenter skal kunne referere til rapporten; uden tabellen kan s\aa{}danne referencer ikke finde sted.
\subsection{L-kode} \label{sec:l}
Logiweb-dokumentet \cite{kn:base} indeholder et funktionelt programmeringssprog, som jeg vil kalde for ``L''. Der er en del forekomster af L-kode i rapporten. For en forklaring af de konstruktioner fra L, som jeg bruger, vil jeg henvise til funktionsbeskrivelserne i appendikset til \cite{kn:check}, afsnit 3.2, s.\ 6. Herudover vil jeg supplere med et par kommentarer undervejs.
\newpage
\section{Syntaks for " [ math system zf end math ] " } \label{sec:syntaks}
Som n\ae{}vnt i indledningen vil jeg arbejde med en reduceret version af ZF m\ae{}ngdel\ae{}re, som jeg kalder for ``" [ math system zf end math ] "''. Dette afsnit beskriver syntaksen for " [ math system zf end math ] ". Der er to syntaktiske hovedkategorier i " [ math system zf end math ] ": Termer og formler. Underafsnit \ref{sec:termer} beskriver syntaksen for termer, og underafsnit \ref{sec:formler} beskriver syntaksen for formler.
\subsection{Termer} \label{sec:termer}
Syntaksen for en term " [ math metavar var t end metavar end math ] " kan beskrives ved den f\o{}lgende BNF-grammatik:
%
\begin{eqnarray*}
" [ metavar var t end metavar ] " & ::= & " [ vaerdi ] " \; | \; " [ variabel ] " \\
" [ vaerdi ] "
& ::= & " [ zermelo empty set ] " \; | \;
" [ zermelo pair metavar var t end metavar comma metavar var t end metavar end pair ] " \; | \;
" [ union metavar var t end metavar end union ] " \; | \;
" [ power metavar var t end metavar end power ] " \; | \;
" [ the set of ph in metavar var t end metavar such that metavar var f end metavar end set ] " \\
" [ variabel ] " & ::= & " [ object-var ] " \; | \; " [ ex-var ] " \; | \;
" [ ph-var ] "
%\label{eq:term}
\end{eqnarray*}
%
En " [ vaerdi ] " svarer til en konkret m\ae{}ngde --- v\ae{}rditermer indeholder ingen variable. Den grundl\ae{}ggende v\ae{}rdi er den tomme m\ae{}ngde; herudfra kan vi konstruere par, f\ae{}llesm\ae{}ngder, potensm\ae{}ngder samt delm\ae{}ngder hvis elementer opfylder en bestemt egenskab. Der er s\aa{}ledes ingen individuelle konstanter i " [ math system zf end math ] "; alt er m\ae{}ngder.
En " [ variabel ] " kan for det f\o{}rste v\ae{}re en objekt-variabel, som varierer over v\ae{}rdier\footnote{Vi repr\ae{}senterer objektvariable med symbolerne ``" [ math object var var a end var end math ] "'', ``" [ math object var var b end var end math ] "'', $\cdots$ ``" [ math object var var z end var end math ] "'' samt ``" [ math object var var big set end var end math ] "''.}. De to andre typer af variable --- eksistens-variable og pladsholder-variable --- vil jeg vente med at forklare til hhv.\ afsnit \ref{sec:eksistens} og \ref{sec:separation}.
\subsection{Formler} \label{sec:formler}
Syntaksen for en formel " [ math metavar var f end metavar end math ] " kan beskrives ved den f\o{}lgende BNF-grammatik:
%
\begin{eqnarray*}
" [ metavar var f end metavar ] " & ::= & " [ metavar var t end metavar zermelo in metavar var t end metavar ] " \; | \;
" [ metavar var t end metavar zermelo is metavar var t end metavar ] " \; | \;
" [ not0 metavar var f end metavar ] " \; | \;
" [ metavar var f end metavar imply metavar var f end metavar ] " \; | \;
\forall \texttt{Objekt-var}
\colon {\cal F}
\end{eqnarray*}
%
\noindent Med en formel kan vi alts\aa{} p\aa{}st\aa{}, at en m\ae{}ngde tilh\o{}rer en anden m\ae{}ngde, eller at to m\ae{}ngder er lig hinanden. Desuden kan vi negere formler\footnote{Jeg skriver negationstegnet med en prik over for at undg\aa{} forveksling med konstruktionen " [ bracket not var x end bracket ] " fra \cite{kn:base}.}, lade formler implicere hinanden, samt kvantificere formler med objektvariable.
\subsection{Objektvariable vs.\ metavariable} \label{sec:objekt}
En metavariabel\footnote{Vi repr\ae{}senterer metavariable med symbolerne ``" [ math metavar var a end metavar end math ] "'', ``" [ math metavar var b end metavar end math ] "'', $\cdots$ ``" [ math metavar var z end metavar end math ] "'' samt ``" [ math metavar var big set end metavar end math ] "''.}
er en variabel, der varierer over vilk\aa{}rlige termer --- alts\aa{} ogs\aa{} over objektvariable. Denne rapport er ikke helt fri for objektvariable; men n\aa{}r vi taler om " [ math system zf end math ] " i aksiomer\footnote{Strengt taget burde jeg skrive ``aksiomskemaer'', da vi bruger metavariable. Da der kun er aksiomskemaer i denne rapport, vil jeg imidlertid bruge ordet ``aksiom'' for at g\o{}re teksten lidt lettere.}, definitioner og beviser vil jeg s\aa{} vidt muligt bruge metavariable, da de er mere fleksible end objektvariable. F.eks.\ kan ``" [ math object var var s end var end math ] "''
i formlen " [ bracket object var var s end var zermelo is metavar var x end metavar end bracket ] "\footnote{En fodnote om stil: Jeg vil ofte indramme matematiske udtryk i firkantede parenteser ``$[\: ]$'' for at adskille udtrykkene fra den omgivende tekst. De firkantede parenteser har ingen selvst\ae{}ndig betydning.} kun instantieres til en term; vi kan ikke skifte objektvariabel og konkludere " [ bracket object var var t end var zermelo is metavar var x end metavar end bracket ] ". Derimod kan vi sagtens instantiere formlen " [ bracket metavar var s end metavar zermelo is metavar var x end metavar end bracket ] " til " [ bracket object var var t end var zermelo is metavar var x end metavar end bracket ] ". En liste over de metavariable, jeg vil bruge, kan ses i bilag \ref{sec:variable}.
%\newpage
\section{Aksiomatisk system} \label{sec:aksiomsystem}
" [ math system zf end math ] " er en teori i 1.\ ordens pr\ae{}dikatkalkyle. Vi kan s\aa{}ledes opdele aksiomsystemet for " [ math system zf end math ] " i to: En pr\ae{}dikatlogisk del og en m\ae{}ngdeteoretisk del. Underafsnit \ref{sec:1order} gennemg\aa{}r den pr\ae{}dikatlogiske del, og underafsnit \ref{sec:axiomzf} gennemg\aa{}r den m\ae{}ngdeteoretiske del. Bilag \ref{sec:helezf} indeholder en kopi af det samlede aksiomsystem.
\subsection{1.\ ordens pr\ae{}dikatkalkyle} \label{sec:1order}
Her er de f\o{}rste slutningsregler i " [ math system zf end math ] ":
\display{
" [ math define statement of system zf as all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar rule plus all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power rule plus all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar rule plus all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var x end metavar zermelo is metavar var y end metavar infer the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot 1deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end 1deduction endorse metavar var a end metavar infer metavar var b end metavar rule plus all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar rule plus all metavar var x end metavar indeed all metavar var t end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed exist-sub0 quote metavar var a end metavar end quote is quote metavar var b end metavar end quote where quote metavar var x end metavar end quote is quote metavar var t end metavar end quote end sub endorse metavar var a end metavar infer metavar var b end metavar rule plus all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union rule plus all metavar var x end metavar indeed all metavar var a end metavar indeed metavar var a end metavar infer for all objects metavar var x end metavar indeed metavar var a end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var z end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var z end metavar end quote end sub endorse not0 metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar infer metavar var b end metavar rule plus all metavar var s end metavar indeed not0 metavar var s end metavar zermelo in zermelo empty set end define end math ] "
" [ math define statement of 1rule mp as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule mp as rule tactic end define end math ] "
" [ math define statement of 1rule gen as system zf infer all metavar var x end metavar indeed all metavar var a end metavar indeed metavar var a end metavar infer for all objects metavar var x end metavar indeed metavar var a end metavar end define , define proof of 1rule gen as rule tactic end define end math ] "
" [ math define statement of 1rule repetition as system zf infer all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end define , define proof of 1rule repetition as rule tactic end define end math ] "
" [ math define statement of 1rule ad absurdum as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule ad absurdum as rule tactic end define end math ] "
" [ math define statement of 1rule deduction as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot 1deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end 1deduction endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule deduction as rule tactic end define end math ] "
}
F\o{}rst et par ord om syntaks: Konstruktionen " [ bracket var x infer var y end bracket ] " st\aa{}r for inferens, alts\aa{} at vi kan bevise " [ math var y end math ] ", hvis vi har et bevis for " [ math var x end math ] ". Konstruktionen " [ bracket var x endorse var y end bracket ] " betyder, at
" [ math var y end math ] " g\ae{}lder, hvis sidebetingelsen " [ math var x end math ] " er sand. Endelig er konstruktionen " [ bracket all var x indeed var y end bracket ] " en meta-alkvantor; betydningen er, at metavariablen " [ math var x end math ] " kan instantieres til hvad som helst --- selv andre metavariable.
De viste slutningsregler er n\ae{}sten identiske med
reglerne " [ math rule mp end math ] ", " [ math rule gen end math ] ", " [ math double negation end math ] " og " [ math deduction end math ] " i systemet " [ math system s end math ] " fra \cite{kn:check}. Der er dog to \ae{}ndringer. For det f\o{}rste har jeg tilf\o{}jet " [ bracket all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end bracket ] " som en slutningsregel. (I \cite{kn:check} vises " [ bracket all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end bracket ] " med et lav-niveau bevis). For det andet har jeg lavet en lille \ae{}ndring i deduktionsreglen " [ math 1rule deduction end math ] ", som g\o{}r den i stand til at h\aa{}ndtere sidebetingelser bedre. Jeg beskriver \ae{}ndringen i bilag \ref{sec:deduktion}.
Som n\ae{}vnt i \cite{kn:check} kan deduktionsreglen erstatte enhver anvendelse af aksiomskemaerne A4 og A5 fra \cite{kn:mendel} (se evt.\ Mendelsons system p\aa{} s.\ 69). Derfor har jeg ikke medtaget disse aksiomskemaer. P\aa{} denne m\aa{}de f\aa{}r vi ogs\aa{} afpr\o{}vet deduktionsreglens brugervenlighed; vi skal straks se et eksempel, hvor A4 kunne have gjort nytte.
\verb| |
\subsubsection{H\aa{}ndtering af eksistenskvantorer}
\label{sec:eksistens}
Et sp\o{}rgsm\aa{}l er nemlig, hvordan man skal h\aa{}ndtere introduktion og elimination af eksistenskvantorer --- alts\aa{} slutninger som f.eks.\
" [ bracket zermelo empty set zermelo is zermelo empty set infer exist object var var x end var indeed object var var x end var zermelo is object var var x end var end bracket ] " og
" [ bracket exist object var var x end var indeed object var var x end var zermelo is object var var x end var infer object var var c end var zermelo is object var var c end var end bracket ] " (hvor ``" [ math object var var c end var end math ] "'' er et ikke tidligere anvendt navn p\aa{} en konstant).
Det er muligt at h\aa{}ndtere denne slags slutninger alene med reglerne fra afsnit \ref{sec:1order}, men det er omst\ae{}ndigt og tidskr\ae{}vende (sammenlign f.eks.\ de to beviser p\aa{} s.\ 81 i \cite{kn:mendel}). Som et minimum kr\ae{}ver det, at man har A4 fra \cite{kn:mendel} til r\aa{}dighed --- og som n\ae{}vnt ovenfor har jeg valgt ikke at medtage dette aksiomskema.
Jeg har i stedet implementeret en l\o{}sning, der er baseret p\aa{} begrebet ``eksistensvariabel''. Vi indf\o{}rer en un\ae{}r operator " [ bracket var x is existential var end bracket ] " og definerer, at en term er en ekistens-variabel, hviss den har " [ bracket var x is existential var end bracket ] " som principal operator. Funktionen " [ bracket var x is existential var end bracket ] " tester, om " [ math var x end math ] " er en eksistens-variabel:
\display{" [ math define value of var x is existential var as var x term root equal quote existential var var x end var end quote end define end math ] "\footnote{Konstruktionen " [ math define value of var x as var y end define end math ] " er en s\aa{}kaldt ``v\ae{}rdidefinition'' i L (jvf.\ afsnit \ref{sec:l}). Den svarer til en almindelig funktionsdefinition; vi knytter funktionssignaturen " [ math var x end math ] " til kroppen " [ math var y end math ] ".}
}
\noindent Vi kan da definere de fire eksistens-variable, som denne rapport vil g\o{}re brug af (jvf.\ bilag \ref{sec:variable}):
\verb| |
\display{
" [ math define macro of ex1 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex1 as existential var var a end var end define end quote end define end define end math ] "\footnote{Konstruktionen " [ math define macro of var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x as var y end define end quote end define end define end math ] " st\aa{}r for ``makrodefinition'' i L. Vi definerer " [ math var x end math ] " som v\ae{}rende en forkortelse for " [ math var y end math ] ". Ud fra bevischeckerens synspunkt er der ingen forskel p\aa{} " [ math var x end math ] " og " [ math var y end math ] ".}
" [ math define macro of ex2 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex2 as existential var var b end var end define end quote end define end define end math ] "
" [ math define macro of ex10 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex10 as existential var var j end var end define end quote end define end define end math ] "
" [ math define macro of ex20 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex20 as existential var var t end var end define end quote end define end define end math ] "\footnote{``j'' og ``t'' er hhv.\ bogstav nr.\ 10 og 20 i alfabetet.}
}
\noindent Ideen med disse definitioner er at repr\ae{}sentere en formel som f.eks.\ \mbox{" [ bracket exist object var var x end var indeed object var var x end var zermelo is object var var x end var end bracket ] "} ved formlen " [ bracket metavar var x end metavar zermelo is metavar var x end metavar end bracket ] ", hvor ``" [ math metavar var x end metavar end math ] "'' er en eksistensvariabel. P\aa{} denne m\aa{}de kommer eksistensvariablene til at fungere som erstatning for eksistenskvantoren. Dette betyder ogs\aa{}, at der ikke er brug for at nogen regel, der eksplicit fjerner eksistenskvantorer --- de er allerede v\ae{}k.
Til geng\ae{}ld f\aa{}r vi brug for en regel, der kan introducere eksistensvariable. Til den ende definerer vi pr\ae{}dikatet " [ math exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " i L:
\verb| |
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define macro of exist-sub var a is var b where var x is var t end sub as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define exist-sub var a is var b where var x is var t end sub as exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end define end quote end define end define end math ] "
\item " [ math define value of exist-sub0 var a is var b where var x is var t end sub as lambda var c dot var x is existential var and exist-sub1 var a is var b where var x is var t end sub end define end math ] "
\item " [ math define value of exist-sub1 var a is var b where var x is var t end sub as var a tagged guard var x tagged guard var t tagged guard newline tagged if var b term root equal quote for all objects var u indeed var v end quote then false else newline tagged if var b is existential var and var b term equal var x then var a term equal var t else tagged if newline var a term root equal var b then exist-sub* var a tail is var b tail where var x is var t end sub else false end if end if end if end define end math ] "
\item " [ math define value of exist-sub* var a is var b where var x is var t end sub as var b tagged guard var x tagged guard var t tagged guard tagged if var a then true else tagged if exist-sub1 var a head is var b head where var x is var t end sub then exist-sub* var a tail is var b tail where var x is var t end sub else false end if end if end define end math ] "
\end{list}
\noindent Pr\ae{}dikatet " [ math exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " er sandt, hvis " [ math var x end math ] " er en eksistens-variabel, og hvis formlen " [ math var a end math ] " er identisk med resultatet af at erstatte alle forekomster af " [ math var x end math ] " i formlen " [ math var b end math ] " med termen " [ math var t end math ] ". F.eks.\ er " [ math exist-sub0 quote zermelo empty set zermelo is zermelo empty set end quote is quote existential var var a end var zermelo is existential var var a end var end quote where quote existential var var a end var end quote is quote zermelo empty set end quote end sub end math ] " sand. Herudover er det et krav, at hverken " [ math var a end math ] " eller " [ math var b end math ] " m\aa{} indeholde objektkvantorer; s\aa{}ledes er " [ math exist-sub0 quote for all objects object var var s end var indeed object var var s end var zermelo is zermelo empty set end quote is quote for all objects object var var s end var indeed object var var s end var zermelo is existential var var a end var end quote where quote existential var var a end var end quote is quote zermelo empty set end quote end sub end math ] " falsk. Dette krav er udelukkende indf\o{}rt for at g\o{}re koden for " [ math exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " s\aa{} simpel som mulig; det har ikke v\ae{}ret n\o{}dvendigt at s\ae{}tte " [ math exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " i stand til at h\aa{}ndtere kvantificering.
Vi kan nu definere den slutningsregel, der st\aa{}r for introduktion af eksistensvariable:
\display{
" [ math define statement of 1rule exist intro as system zf infer all metavar var x end metavar indeed all metavar var t end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed exist-sub0 quote metavar var a end metavar end quote is quote metavar var b end metavar end quote where quote metavar var x end metavar end quote is quote metavar var t end metavar end quote end sub endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule exist intro as rule tactic end define end math ] "
%
\footnote{I denne definition varierer ``" [ math metavar var x end metavar end math ] "'' over eksistens-variable.}
%
}
\noindent Med " [ math 1rule exist intro end math ] " kan vi f.eks.\ slutte " [ bracket existential var var a end var zermelo is existential var var a end var end bracket ] " ud fra " [ bracket zermelo empty set zermelo is zermelo empty set end bracket ] ". Vi har nu defineret seks aksiomer, der tilsammen d\ae{}kker 1. ordens pr\ae{}dikatkalkyle.
\verb| |
\subsection{Selve m\ae{}ngdel\ae{}ren} \label{sec:axiomzf}
De aksiomer i " [ math system zf end math ] ", der vedr\o{}rer selve m\ae{}ngdel\ae{}ren, har jeg hentet fra kapitel 4.3 og 4.4 i \cite{kn:gold}. Her er de f\o{}rste fem:
\display{
" [ math define statement of axiom extensionality as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end define , define proof of axiom extensionality as rule tactic end define end math ] "
" [ math define statement of axiom empty set as system zf infer all metavar var s end metavar indeed not0 metavar var s end metavar zermelo in zermelo empty set end define , define proof of axiom empty set as rule tactic end define end math ] "
" [ math define statement of axiom pair definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end define , define proof of axiom pair definition as rule tactic end define end math ] "
" [ math define statement of axiom union definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union end define , define proof of axiom union definition as rule tactic end define end math ] "
" [ math define statement of axiom power definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power end define , define proof of axiom power definition as rule tactic end define end math ] "\footnote{Konstruktionerne " [ bracket not0 var x imply not0 var y end bracket ] ", " [ bracket not0 var x imply var y end bracket ] " og " [ bracket not0 var x imply var y imply not0 var y imply var x end bracket ] " svarer til de kendte konnektiver; de bliver defineret formelt i afsnit \ref{sec:konnektiver}.}
}
Reglen " [ math axiom extensionality end math ] " siger, at to m\ae{}ngder er ens, hviss de har de samme elementer. De \o{}vrige fire regler definerer begreberne ``tom m\ae{}ngde'', ``par'', ``foreningssm\ae{}ngde'' og ``potensm\ae{}ngde''. L\ae{}g m\ae{}rke til, at
" [ bracket union metavar var t end metavar end union end bracket ] " er en un\ae{}r operator; ideen er, at " [ math union metavar var t end metavar end union end math ] " er lig med foreningsm\ae{}ngden af alle de m\ae{}ngder, som " [ math metavar var t end metavar end math ] " indeholder.
Bem\ae{}rk ogs\aa{} at to af aksiomerne indeholder objektvariablen " [ math object var var s end var end math ] ". Forklaringen herp\aa{} er, at aksiomerne indeholder objektkvantoren " [ bracket for all objects var x indeed var y end bracket ] "; og deduktionsreglen fra \cite{kn:check} er ikke egnet til at h\aa{}ndtere kombinationen ``objektkvantor og metavariabel''. Derfor vil " [ math var x end math ] " i " [ bracket for all objects var x indeed var y end bracket ] " altid v\ae{}re en objektvariabel i denne rapport.
\verb| |
\subsubsection{Separation og pladsholdervariable}
\label{sec:separation}
Vi mangler stadigv\ae{}k et ``separationsaksiom'' --- dvs.\ et aksiom, der giver mening til konstruktionen " [ bracket the set of ph in metavar var t end metavar such that metavar var f end metavar end set end bracket ] " fra afsnit \ref{sec:termer}. For at implementere separationsaksiomet indf\o{}rer vi begrebet ``pladsholder-variabel''. Fremgangsm\aa{}den er den samme som ved eksistens-variable i afsnit \ref{sec:eksistens}. F\o{}rst indf\o{}rer vi en un\ae{}r operator " [ bracket placeholder-var var x end var end bracket ] " og definerer, at en term er en pladsholder-variabel, hviss den har " [ bracket placeholder-var var x end var end bracket ] " som prim\ae{}r operator. Funktionen " [ bracket var x is placeholder-var end bracket ] " checker, om " [ math var x end math ] " er en pladsholder-variabel:
%\verb| |
\display{" [ math define value of var x is placeholder-var as var x term root equal quote placeholder-var var x end var end quote end define end math ] "
}
\noindent Vi kan nu definere de pladsholder-variable, vi f\aa{}r brug for, som f\o{}lger (jvf.\ bilag \ref{sec:variable}):
\verb| |
\display{
" [ math define macro of placeholder-var1 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define placeholder-var1 as placeholder-var var a end var end define end quote end define end define end math ] "
" [ math define macro of placeholder-var2 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define placeholder-var2 as placeholder-var var b end var end define end quote end define end define end math ] "
" [ math define macro of placeholder-var3 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define placeholder-var3 as placeholder-var var c end var end define end quote end define end define end math ] "
}
\noindent S\aa{} definerer vi pr\ae{}dikatet " [ math ph-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " i L:
\verb| |
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item
" [ math define macro of ph-sub var a is var b where var x is var t end sub as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ph-sub var a is var b where var x is var t end sub as ph-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end define end quote end define end define end math ] "
\item " [ math define value of ph-sub0 var a is var b where var x is var t end sub as lambda var c dot var x is placeholder-var and ph-sub1 var a is var b where var x is var t end sub end define end math ] "
\item " [ math define value of ph-sub1 var a is var b where var x is var t end sub as var a tagged guard var x tagged guard var t tagged guard newline tagged if var b term root equal quote for all objects var u indeed var v end quote then false else newline tagged if var b is placeholder-var and var b term equal var x then var a term equal var t else newline tagged if var b is existential var then var a term root equal var b else tagged if newline var a term root equal var b then ph-sub* var a tail is var b tail where var x is var t end sub else false end if end if end if end if end define end math ] "
\item " [ math define value of ph-sub* var a is var b where var x is var t end sub as var b tagged guard var x tagged guard var t tagged guard tagged if var a then true else tagged if ph-sub1 var a head is var b head where var x is var t end sub then ph-sub* var a tail is var b tail where var x is var t end sub else false end if end if end define end math ] "
\end{list}
\noindent Pr\ae{}dikatet " [ math ph-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " er sandt, hvis " [ math var x end math ] " er en pladsholder-variabel, og hvis formlen " [ math var a end math ] " er identisk med resultatet af at erstatte alle forekomster af " [ math var x end math ] " i formlen " [ math var b end math ] " med termen " [ math var t end math ] ".
Ligesom ved " [ math exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] " kr\ae{}ver vi, at hverken " [ math var a end math ] " eller " [ math var b end math ] " indeholder objektkvantorer.
F.eks.\ er " [ math ph-sub0 quote zermelo empty set zermelo in metavar var s end metavar end quote is quote placeholder-var var a end var zermelo in metavar var s end metavar end quote where quote placeholder-var var a end var end quote is quote zermelo empty set end quote end sub end math ] " sand. Det er i \o{}vrigt tilladt, at " [ math var a end math ] " og " [ math var b end math ] " indeholder forskellige ekistensvariable; s\aa{}ledes er " [ math ph-sub0 quote existential var var a end var end quote is quote existential var var b end var end quote where quote placeholder-var var a end var end quote is quote zermelo empty set end quote end sub end math ] " sand. P\aa{} denne m\aa{}de kan vi im\o{}dekomme kravet om, at en eksistens-variabel skal v\ae{}re ubrugt, n\aa{}r den introduceres i et bevis.
Vi definerer nu separationsaksiomet " [ math axiom separation definition end math ] " ud fra " [ math ph-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end math ] ":
\display{
" [ math define statement of axiom separation definition as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var z end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var z end metavar end quote end sub endorse not0 metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set end define , define proof of axiom separation definition as rule tactic end define end math ] "
}
\noindent Ved f\o{}rste \o{}jekast virker konjunkten " [ bracket metavar var p end metavar is placeholder-var end bracket ] " i denne definition overfl\o{}dig, fordi " [ math ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var z end metavar end quote end sub end math ] " i forvejen kr\ae{}ver, at " [ math metavar var p end metavar end math ] " er en pladsholder-variabel. Forklaringen er, at ``" [ math metavar var p end metavar end math ] "'' kun optr\ae{}der i definitionens sidebetingelse. Dette betyder, at bevischeckeren har sv\ae{}rt ved at finde ud af, hvordan ``" [ math metavar var p end metavar end math ] "'' skal instantieres, n\aa{}r vi anvender definitionen. Det ekstra krav " [ bracket metavar var p end metavar is placeholder-var end bracket ] " hj\ae{}lper bevischeckeren til at instantiere ``" [ math metavar var p end metavar end math ] "'' korrekt.
Med " [ math axiom separation definition end math ] " til r\aa{}dighed kan vi nu f.eks.\ definere " [ math the set of ph in var x such that zermelo empty set zermelo in placeholder-var var a end var end set end math ] " som den delm\ae{}ngde af " [ math var x end math ] ", hvis elementer indeholder " [ math zermelo empty set end math ] ":
\verb| |
\display{" [ math define macro of contains-empty var x end empty as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define contains-empty var x end empty as the set of ph in var x such that zermelo empty set zermelo in placeholder-var1 end set end define end quote end define end define end math ] ",}
\noindent og vi kan slutte " [ bracket not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 zermelo empty set zermelo in metavar var s end metavar end bracket ] " ud fra " [ bracket metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that zermelo empty set zermelo in placeholder-var var a end var end set end bracket ] ".
Vi bruger alts\aa{} en pladsholder-variabel som fri variabel i d\'{e}n formel " [ math metavar var f end metavar end math ] ", der definerer delm\ae{}ngden " [ math the set of ph in metavar var x end metavar such that metavar var f end metavar end set end math ] ". Ideen med at bruge pladsholder-variable (frem for objekt- eller metavariable) til dette form\aa{}l er, at vi ikke \o{}nsker at kvantificere eller instantiere den frie variabel i " [ math metavar var f end metavar end math ] ". Det er f.eks.\ noget sludder at skrive ``" [ math for all objects placeholder-var var a end var indeed the set of ph in metavar var x end metavar such that zermelo empty set zermelo in placeholder-var var a end var end set end math ] "'' eller at konkludere ``" [ math the set of ph in metavar var x end metavar such that zermelo empty set zermelo in zermelo empty set end set end math ] "'' ud fra definitionen af " [ math the set of ph in metavar var x end metavar such that zermelo empty set zermelo in placeholder-var var a end var end set end math ] ". Den frie variabel i " [ math metavar var f end metavar end math ] " skal blot v\ae{}re en pladsholder; derfor pladsholder-variable.
\verb| |
\section{Makrodefinitioner} \label{sec:makro}
Dette afsnit indeholder d\'{e} makrodefinitioner, som vi vil g\o{}re brug af i resten af rapporten. Definitionerne drejer sig for det meste om m\ae{}ngdeteoretiske begreber, f.eks.\ ``\ae{}kvivalensklasse'' og ``partition''. Til sidst i afsnittet formulerer vi hovedresultatet --- at der til enhver \ae{}kvivalensrelation svarer en partition --- som et formelt teorem.
\subsection{Konnektiver} \label{sec:konnektiver}
Ud fra de to basale konnektiver " [ bracket not0 var x end bracket ] " og " [ bracket var x imply var y end bracket ] " definerer vi konjunktion, disjunktion og dobbeltimplikation:
\display{" [ math define macro of var x and0 var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x and0 var y as not0 parenthesis var x imply not0 var y end parenthesis end define end quote end define end define end math ] "
" [ math define macro of var x or0 var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x or0 var y as not0 var x imply var y end define end quote end define end define end math ] "
" [ math define macro of var x iff var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x iff var y as parenthesis var x imply var y end parenthesis and0 parenthesis var y imply var x end parenthesis end define end quote end define end define end math ] "
}
\subsection{Negerede formler}
Det er ganske enkelt at definere negeret lighed (" [ math not0 var x zermelo is var y end math ] ") og negeret medlemskab (" [ math not0 var x zermelo in var y end math ] "):
\display{
" [ math define macro of var x zermelo ~is var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x zermelo ~is var y as not0 var x zermelo is var y end define end quote end define end define end math ] "
" [ math define macro of var x zermelo ~in var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x zermelo ~in var y as not0 var x zermelo in var y end define end quote end define end define end math ] "\footnote{H\o{}jresiderne i disse definitioner skal l\ae{}ses som hhv.\ " [ bracket not0 var x zermelo is var y end bracket ] " og " [ bracket not0 var x zermelo in var y end bracket ] ".}
}
\subsection{Delm\ae{}ngde} \label{sec:subset}
M\ae{}ngden " [ math var x end math ] " er en delm\ae{}ngde af " [ math var y end math ] " hviss ethvert medlem af " [ math var x end math ] " ogs\aa{} tilh\o{}rer " [ math var y end math ] ":
\display{
" [ math define macro of var x is subset of var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x is subset of var y as parenthesis object s zermelo in var x imply object s zermelo in var y end parenthesis end define end quote end define end define end math ] "
}
Vi kommer ikke til at bruge denne definition ret ofte. Man f\aa{}r tit en bedre f\o{}ling med, hvad der foreg\aa{}r i beviserne, hvis man skriver definitionen ud. Desuden bruger denne definition af " [ bracket object var var s end var zermelo in var x imply object var var s end var zermelo in var y end bracket ] " objektvariable og implikation; vi vil ofte foretr\ae{}kke at bruge metavariable og inferens i stedet (som f.eks.\ i \mbox{" [ bracket metavar var s end metavar zermelo in var x infer metavar var s end metavar zermelo in var y end bracket ] "}).
\verb| |
\subsection{Singleton-m\ae{}ngde} \label{sec:single}
" [ bracket zermelo pair var x comma var x end pair end bracket ] " er m\ae{}ngden, der indeholder " [ math var x end math ] " som sit eneste
element. Vi definerer " [ bracket zermelo pair var x comma var x end pair end bracket ] " ved at parre " [ math var x end math ] " med sig selv:
\display{" [ math define macro of zermelo singleton var x end singleton as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define zermelo singleton var x end singleton as zermelo pair var x comma var x end pair end define end quote end define end define end math ] "}
\subsection{Bin\ae{}r foreningsm\ae{}ngde og f\ae{}llesm\ae{}ngde}
\label{sec:makrobin}
Vi definerer foreningsm\ae{}ngden mellem to m\ae{}ngder " [ math var x end math ] " og " [ math var y end math ] " som f\o{}lger:
\display{
" [ math define macro of binary-union var x comma var y end union as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define binary-union var x comma var y end union as union zermelo pair zermelo singleton var x end singleton comma zermelo singleton var y end singleton end pair end union end define end quote end define end define end math ] "
}
F\ae{}llesm\ae{}ngden mellem to m\ae{}ngder " [ math var x end math ] " og " [ math var y end math ] " er en delm\ae{}ngde af deres foreningsm\ae{}ngde:
\verb| |
\display{
" [ math define macro of intersection var x comma var y end intersection as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define intersection var x comma var y end intersection as the set of ph in binary-union var x comma var y end union such that placeholder-var3 zermelo in var x and0 placeholder-var3 zermelo in var y end set end define end quote end define end define end math ] "
}
\subsection{Relation} \label{sec:relation}
Det ordnede par " [ math zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair end pair end math ] " indeholder " [ math var x end math ] " som ``f\o{}rstekomponent'' og " [ math var y end math ] " som ``andenkomponent''. Den f\o{}lgende definition af " [ math zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair end pair end math ] " er den mest udbredte i litteraturen (se f.eks.\ afsnit 4.3 i \cite{kn:gold} og afsnit 2.1 i \cite{kn:hrba}):
\verb| |
\display{" [ math define macro of zermelo ordered pair var x comma var y end pair as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define zermelo ordered pair var x comma var y end pair as zermelo pair zermelo singleton var x end singleton comma zermelo pair var x comma var y end pair end pair end define end quote end define end define end math ] "}
Vi kan nu definere en ``relation'' som en m\ae{}ngde af ordnede par. Vi udtrykker denne definition ved at formalisere, hvad det vil sige, at " [ math var x end math ] " er relateret til " [ math var y end math ] " i kraft af relationen " [ math var r end math ] ":
\verb| |
\display{" [ math define macro of var x is related to var y under var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x is related to var y under var r as zermelo ordered pair var x comma var y end pair zermelo in var r end define end quote end define end define end math ] "}
Vi kommer faktisk ikke til at bruge disse to definitioner i rapporten; beviserne vil behandle " [ bracket zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair end pair zermelo in var r end bracket ] " som en primitiv konstruktion. Men det er alligevel betryggende at have det formelle grundlag for relationsbegrebet p\aa{} plads.
\newpage
\subsection{\AE{}kvivalensrelation} \label{sec:eqrel}
At en relation er refleksiv p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " er relateret til sig selv:
\display{" [ math define macro of var r is reflexive relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var r is reflexive relation in var x as for all object s indeed parenthesis object s zermelo in var x imply object s is related to object s under var r end parenthesis end define end quote end define end define end math ] "}
At en relation er symmetrisk p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " opfylder den f\o{}lgende implikation:
\verb| |
\display{" [ math define macro of var r is symmetric relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var r is symmetric relation in var x as for all object s comma object t indeed parenthesis object s zermelo in var x imply object t zermelo in var x imply object s is related to object t under var r imply object t is related to object s under var r end parenthesis end define end quote end define end define end math ] "}
At en relation er transitiv p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " opfylder den f\o{}lgende implikation:
\verb| |
\display{" [ math define macro of var r is transitive relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var r is transitive relation in var x as macro newline for all object s comma object t comma object u indeed parenthesis object s zermelo in var x imply object t zermelo in var x imply object u zermelo in var x imply object s is related to object t under var r imply object t is related to object u under var r imply object s is related to object u under var r end parenthesis end define end quote end define end define end math ] "}
Endelig er en \ae{}kvivalensrelation det samme som en relation, der er refleksiv, symmetrisk og transitiv:
\verb| |
\display{" [ math define macro of var r is equivalence relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var r is equivalence relation in var x as var r is reflexive relation in var x and0 var r is symmetric relation in var x and0 var r is transitive relation in var x end define end quote end define end define end math ] "}
\subsection{M\ae{}ngde-variable}
Mange af rapportens beviser sker i forhold til en uspecificeret m\ae{}ngde. Vi vil referere til denne m\ae{}ngde med metavariablen " [ math metavar var big set end metavar end math ] " og objektvariablen " [ math object var var big set end var end math ] ":
\display{
" [ math define macro of meta big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta big set as metavar var big set end metavar end define end quote end define end define end math ] "
" [ math define macro of object big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define object big set as object var var big set end var end define end quote end define end define end math ] "\footnote{Navnene ``" [ math metavar var big set end metavar end math ] "'' og ``" [ math object var var big set end var end math ] "'' st\aa{}r for hhv.\ for ``big set'' og ``object big set''. Konstruktionerne " [ bracket metavar var x end metavar end bracket ] " og " [ bracket object var var x end var end bracket ] " omdanner " [ math var x end math ] " til hhv.\ en meta- og en objektvariabel. Variablen " [ bracket var big set end bracket ] " vil ogs\aa{} blive brugt i nogle af de kommende definitioner, men ikke i selve beviserne.}
}
\noindent Vi vil s\aa{} vidt muligt bruge metavariablen, men i afsnit \ref{sec:sameinter} og senere bliver det n\o{}dvendigt at g\aa{} over til objektvariablen.
\verb| |
\subsection{\AE{}kvivalensklasse} \label{sec:eqclass}
Lad " [ math var r end math ] " v\ae{}re en \ae{}kvivalensrelation defineret p\aa{} " [ math var big set end math ] ", og lad " [ math var x end math ] " v\ae{}re et medlem af " [ math var big set end math ] ". Vi definerer \ae{}kvivalensklassen " [ math the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma var x end pair end pair zermelo in var r end set end math ] " som den delm\ae{}ngde af " [ math var big set end math ] ", hvis medlemmer st\aa{}r i forhold til " [ math var x end math ] ":
\display{
" [ math define macro of equivalence class of var x in var big set modulo var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define equivalence class of var x in var big set modulo var r as the set of ph in var big set such that placeholder-var1 is related to var x under var r end set end define end quote end define end define end math ] "
}
\AE{}kvivalenssystemet " [ math the set of ph in power var big set end power such that not0 existential var var t end var zermelo in var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in var r end set zermelo is placeholder-var var b end var end set end math ] " er m\ae{}ngden af alle de \ae{}kvivalensklasser, som " [ math var big set end math ] " definerer p\aa{} " [ math var r end math ] ". Vi definerer " [ math the set of ph in power var big set end power such that not0 existential var var t end var zermelo in var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in var r end set zermelo is placeholder-var var b end var end set end math ] " som en delm\ae{}ngde af potensm\ae{}ngden " [ math power var big set end power end math ] ":
\verb| |
\display{
" [ math define macro of eq-system of var big set modulo var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define eq-system of var big set modulo var r as the set of ph in power var big set end power such that ex20 zermelo in var big set and0 equivalence class of ex20 in var big set modulo var r zermelo is placeholder-var2 end set end define end quote end define end define end math ] "
}
\newpage
\subsection{Partition} \label{sec:partition}
En partition af en m\ae{}ngde " [ math var big set end math ] " er en m\ae{}ngde " [ math var p end math ] ", som opfylder tre krav:
%
\begin{enumerate}
\item Ingen af m\ae{}ngderne i " [ math var p end math ] " er tomme.
\item Alle m\ae{}ngderne i " [ math var p end math ] " er indbyrdes disjunkte.
\item Foreningsm\ae{}ngden af alle m\ae{}ngderne i " [ math var p end math ] " er lig med " [ math var big set end math ] ".
\end{enumerate}
%
Den formelle version af denne definition ser s\aa{}ledes ud:
\display{
" [ math define macro of var p is partition of var big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var p is partition of var big set as parenthesis for all object s indeed parenthesis object s zermelo in var p imply object s zermelo ~is zermelo empty set end parenthesis end parenthesis and0 macro newline parenthesis for all object s comma object t indeed parenthesis object s zermelo in var p imply object t zermelo in var p imply object s zermelo ~is object t imply intersection object s comma object t end intersection zermelo is zermelo empty set end parenthesis end parenthesis and0 macro newline union var p end union zermelo is var big set end define end quote end define end define end math ] "
}
Vi kan nu formulere hovedresultatet som et formelt lemma (hvor vi bruger objektvariable):
\verb| |
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of theorem eq-system is partition as system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var end define end math ] "
\end{list}
\noindent Form\aa{}let med resten af rapporten er at bevise " [ math theorem eq-system is partition end math ] ".
\section{Udsagnslogisk bibliotek} \label{sec:udsagn}
I dette afsnit vil jeg bevise en samling af udsagnslogiske sandheder ( eller ``tautologier''), som vil blive brugt i de f\o{}lgende afsnit. De fleste af disse tautologier har mange andre anvendelser end lige netop m\ae{}ngdel\ae{}re. Beviserne er fordelt p\aa{} syv underafsnit; figur 1 giver et overblik over, hvordan beviserne forholder sig til hinanden. Jeg vil kommentere de fleste af beviserne; dog er nogle af dem s\aa{} tekniske, at jeg har ladet dem st\aa{} alene.
% figur 1
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,30cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/prop.eps}}
\caption[Bevisstrukturen for tautologierne]{Bevisstrukturen for tautologierne. En pil fra lemma " [ math var x end math ] " til lemma " [ math var y end math ] " betyder, at " [ math var x end math ] " bruges i beviset for " [ math var y end math ] ". MP-lemmaerne fra afsnit \ref{sec:mplemmaer} er ikke vist. ``ImpTrans'' st\aa{}r for ``ImplyTransitivity''.}
\end{figure}
\clearpage
\subsection{MP-lemmaer} \label{sec:mplemmaer}
Man f\aa{}r ofte brug for at anvende slutningsreglen " [ math 1rule mp end math ] " flere gange i tr\ae{}k. Derfor vil jeg begynde med at vise fire lemmaer, der kan klare mellem 2 og 5 anvendelser af " [ math 1rule mp end math ] "\footnote{I afsnit \ref{sec:transdef} f\aa{}r vi faktisk brug for at anvende " [ math 1rule mp end math ] " 6 gange i tr\ae{}k; men et eller andet sted skal man jo stoppe.}:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp2 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar end define end math ] "
\item " [ math define statement of prop lemma mp3 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar end define end math ] "
\item " [ math define statement of prop lemma mp4 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar end define end math ] "
\item " [ math define statement of prop lemma mp5 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer metavar var f end metavar end define end math ] "
\end{list}
\subsubsection{Det f\o{}rste bevis}
Vi begynder med at bevise " [ math prop lemma mp2 end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp2 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar end define end math ] "
\item " [ math define proof of prop lemma mp2 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar imply metavar var c end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent Da dette er rapportens f\o{}rste bevis, vil jeg bringe nogle ekstra kommentarer\footnote{Denne beskrivelse er en revideret udgave af afsnit 5.1 i \cite{kn:peano}.}. Oven over beviset har jeg gentaget definitionen af det, der skal bevises; dette er kun for overblikkets skyld --- det er ikke en formel n\o{}dvendighed. Selve beviset for " [ math prop lemma mp2 end math ] " best\aa{}r af seks linier, nummereret fra 1 til 6. En bevislinie kan have to former. Den f\o{}rste form er:
%
\begin{eqnarray*}
\texttt{Argumentation} & \gg & \texttt{Konklusion} %\label{eq:linie}
\end{eqnarray*}
%
hvor \texttt{Konklusion} er det som linien beviser, mens teksten i \texttt{Argumentation} udg\o{}r en begrundelse for, at \texttt{Konklusion} g\ae{}lder. F.eks.\ siger linie 5, at meta-formlen \mbox{" [ bracket metavar var b end metavar imply metavar var c end metavar end bracket ] "} g\ae{}lder, fordi den kan udledes fra slutningsreglen " [ math 1rule mp end math ] " ved substitution. Argumentationen skal l\ae{}ses p\aa{} den m\aa{}de, at konklusionerne fra linie 2 og 3 bliver brugt som pr\ae{}misser til " [ math 1rule mp end math ] ". Den generelle betydning af konstruktionen " [ bracket var x modus ponens var y end bracket ] " er, at konklusionen fra linie " [ math var y end math ] " bliver brugt som pr\ae{}mis i forhold til " [ math var x end math ] ".
Den anden form, en bevislinie kan have, er:
%
\begin{eqnarray*}
\texttt{N\o{}gleord} & \gg & \texttt{Konklusion} %\label{eq:linie2}
\end{eqnarray*}
%
hvor \texttt{N\o{}gleord} er et af de tre ord ``Arbitrary'', ``Premise'' eller ``Side-condition''. Betydningen af ordene ``Premise'' og ``Side-condition'' er \aa{}benlys: De angiver, at liniens konklusion indg\aa{}r som en pr\ae{}mis (hhv.\ sidebetingelse) i den s\ae{}tning, der skal bevises. F.eks.\ siger bevisets linie 2, at " [ math prop lemma mp2 end math ] " bruger meta-formlen \\ \mbox{" [ bracket metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end bracket ] "}
som pr\ae{}mis. N\aa{}r ordet ``Arbitrary'' bruges, best\aa{}r konklusionen af en liste af meta-variable (f.eks. " [ bracket metavar var a end metavar comma metavar var b end metavar comma metavar var c end metavar end bracket ] " i linie 1). Ideen hermed er at udtrykke, at vi ikke antager noget om de p\aa{}g\ae{}ldende meta-variable, og at vi derfor har ret til at binde dem med en meta-alkvantor i den s\ae{}tning, der skal bevises. I det forh\aa{}ndenv\ae{}rende bevis berettiger linien med ``Arbitrary'' alts\aa{}, at " [ math prop lemma mp2 end math ] " er kvantificeret med " [ bracket all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed cdots end bracket ] ".
Alt dette har drejet sig om den formelle syntaks for et Logiweb bevis. Der er ikke s\aa{} meget at sige om selve beviset for " [ math prop lemma mp2 end math ] "; vi indkapsler simpelthen to p\aa{} hinanden f\o{}lgende anvendelser af " [ math 1rule mp end math ] ".
\subsubsection{Beviser for de andre MP-lemmaer}
Beviserne for de \o{}vrige MP-lemmaer er lige ud ad landevejen:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp3 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar end define end math ] "
\item
" [ math define proof of prop lemma mp3 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar imply metavar var d end metavar cut 1rule mp modus ponens metavar var c end metavar imply metavar var d end metavar modus ponens metavar var c end metavar conclude metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma mp4 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar end define end math ] "
" [ math define proof of prop lemma mp4 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar cut prop lemma mp2 modus ponens metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var c end metavar modus ponens metavar var d end metavar conclude metavar var e end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma mp5 as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer metavar var f end metavar end define end math ] "
" [ math define proof of prop lemma mp5 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer prop lemma mp3 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar modus ponens metavar var c end metavar conclude metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar cut prop lemma mp2 modus ponens metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar modus ponens metavar var d end metavar modus ponens metavar var e end metavar conclude metavar var f end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{ Implikation }
Dette afsnit indeholder en r\ae{}kke lemmaer vedr.\ implikation, grupperet i fire under-underafsnit.
\subsubsection{Refleksivitet; blok-konstruktionen}
Lemmaet " [ math prop lemma auto imply end math ] " udsiger, at implikations-relationen er refleksiv:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma auto imply as system zf infer all metavar var a end metavar indeed metavar var a end metavar imply metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma auto imply as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed metavar var a end metavar infer 1rule repetition modus ponens metavar var a end metavar conclude metavar var a end metavar cut all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar conclude metavar var a end metavar imply metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item Beviset for " [ math prop lemma auto imply end math ] " indeholder to nye ting i forhold til de hidtidige beviser: En bevisblok, og en anvendelse af deduktions-reglen. En bevisblok er selvst\ae{}ndig enhed i et bevis; den afh\ae{}nger ikke af den \o{}vrige del af beviset. Den ovenst\aa{}ende bevisblok indeholder et bevis for lemmaet " [ bracket all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end bracket ] ". Pointen er nu, at blokkens sidste linie (linie 5) fungerer som en forkortelse for dette lemma. Vi kan da anvende deduktionsreglen p\aa{} denne linie til at omdanne inferensen " [ bracket all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end bracket ] " til implikationen " [ bracket metavar var a end metavar imply metavar var a end metavar end bracket ] ". Det vigtigste form\aa{}l med deduktionsreglen er netop, at vi let kan skifte fra inferens til implikation.
\end{list}
\subsubsection{Transitivitet}
Lemmaet " [ math prop lemma imply transitivity end math ] " udsiger, at implikations-relationen er transitiv:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma imply transitivity as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar imply metavar var c end metavar end define end math ] "
\item Vi viser " [ math prop lemma imply transitivity end math ] " ved hj\ae{}lp af " [ math 1rule mp end math ] " og deduktionsreglen:
\item " [ math define proof of prop lemma imply transitivity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var c end metavar conclude metavar var a end metavar imply metavar var b end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply metavar var c end metavar cut prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply metavar var c end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var b end metavar imply metavar var c end metavar conclude metavar var a end metavar imply metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Sv\ae{}kkelse}
Vi f\aa{}r ofte brug for det f\o{}lgende r\ae{}sonnement: Hvis formlen " [ math metavar var a end metavar end math ] " g\ae{}lder ubetinget, s\aa{} g\ae{}lder den ogs\aa{} under antagelse af en vilk\aa{}rlig anden formel " [ math metavar var b end metavar end math ] ". Lemmaet " [ math prop lemma weakening end math ] " udtrykker dette r\ae{}sonnement som f\o{}lger:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma weakening as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item Vi beviser " [ math prop lemma weakening end math ] " ved hj\ae{}lp af deduktionsreglen:
\item " [ math define proof of prop lemma weakening as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar infer 1rule repetition modus ponens metavar var b end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar conclude metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar cut metavar var b end metavar infer 1rule mp modus ponens metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar modus ponens metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Modsigelse}
Det sidste lemma i dette afsnit vedr\o{}rer strengt taget ikke implikation, men derimod inferens (" [ math var x infer var y end math ] "). Lemmaet " [ math prop lemma from contradiction end math ] " udsiger, at vi kan bevise hvad som helst, hvis vi har bevist to formler, der modsiger hinanden:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma from contradiction as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "
\item Beviset bruger " [ math prop lemma weakening end math ] " og slutningsreglen " [ math 1rule ad absurdum end math ] ":
\item " [ math define proof of prop lemma from contradiction as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer prop lemma weakening modus ponens metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar cut prop lemma weakening modus ponens not0 metavar var a end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule ad absurdum modus ponens not0 metavar var b end metavar imply metavar var a end metavar modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{H\aa{}ndtering af dobbeltnegationer}
De to lemmaer " [ math prop lemma remove double neg end math ] " og " [ math prop lemma add double neg end math ] " tillader os hhv.\ at fjerne og tilf\o{}je dobbeltnegationer. Jeg vil ikke kommentere beviserne:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma remove double neg as system zf infer all metavar var a end metavar indeed not0 not0 metavar var a end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma remove double neg as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed not0 not0 metavar var a end metavar infer prop lemma weakening modus ponens not0 not0 metavar var a end metavar conclude not0 metavar var a end metavar imply not0 not0 metavar var a end metavar cut prop lemma auto imply conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut 1rule ad absurdum modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar modus ponens not0 metavar var a end metavar imply not0 not0 metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop lemma add double neg as system zf infer all metavar var a end metavar indeed metavar var a end metavar infer not0 not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma add double neg as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed not0 not0 not0 metavar var a end metavar infer prop lemma remove double neg modus ponens not0 not0 not0 metavar var a end metavar conclude not0 metavar var a end metavar cut all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed not0 not0 not0 metavar var a end metavar infer not0 metavar var a end metavar conclude not0 not0 not0 metavar var a end metavar imply not0 metavar var a end metavar cut metavar var a end metavar infer prop lemma weakening modus ponens metavar var a end metavar conclude not0 not0 not0 metavar var a end metavar imply metavar var a end metavar cut 1rule ad absurdum modus ponens not0 not0 not0 metavar var a end metavar imply metavar var a end metavar modus ponens not0 not0 not0 metavar var a end metavar imply not0 metavar var a end metavar conclude not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Modus tollens og besl\ae{}gtede lemmaer}
Hovedresultatet fra dette afsnit er slutningsreglen modus tollens, bevist som et lemma:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mt as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar end define end math ] "
\item For at vise " [ math prop lemma mt end math ] " begynder vi med et teknisk lemma, der ikke har den store v\ae{}rdi i sig selv:
\item " [ math define statement of prop lemma technicality as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma technicality as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar infer prop lemma remove double neg modus ponens not0 not0 metavar var a end metavar conclude metavar var a end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\item Uafh\ae{}ngigt af " [ math prop lemma technicality end math ] " kan vi vise en version af " [ math prop lemma mt end math ] ", hvor " [ math metavar var a end metavar end math ] " optr\ae{}der i negeret form:
\item " [ math define statement of prop lemma negative mt as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma negative mt as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma weakening modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut 1rule ad absurdum modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item Ud fra " [ math prop lemma technicality end math ] " og " [ math prop lemma negative mt end math ] " kan vi nu vise " [ math prop lemma mt end math ] ":
\item " [ math define statement of prop lemma mt as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma mt as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma technicality conclude not0 not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma negative mt modus ponens not0 not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item Vi slutter dette underafsnit med en variant af " [ math prop lemma mt end math ] ", som erstatter en inferens med en implikation:
\item " [ math define statement of prop lemma contrapositive as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar end define end math ] "
\item N\aa{}r en inferens skal erstattes med en implikation, er det altid deduktionsreglen, der skal i spil:
\item " [ math define proof of prop lemma contrapositive as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma mt modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply not0 metavar var a end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Konjunktion}
Hovedm\aa{}let med dette underafsnit er at konvertere mellem formlerne " [ math metavar var a end metavar end math ] " og " [ math metavar var b end metavar end math ] " og deres konjunktion " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ".
\subsubsection{Forening af konjunkter}
Vi begynder med at sl\aa{} " [ math metavar var a end metavar end math ] " og " [ math metavar var b end metavar end math ] " sammen til " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma join conjuncts as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var b end metavar infer not0 metavar var a end metavar imply not0 metavar var b end metavar end define end math ] "
\item Beviset for " [ math prop lemma join conjuncts end math ] " er af teknisk karakter. Vi viser den makroekspanderede form " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ", som vi i bevisets sidste linie konverterer til " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ". Denne sidste linie er ikke n\o{}dvendig for bevischeckeren, men den g\o{}r beviset lidt nemmere at l\ae{}se:
\item " [ math define proof of prop lemma join conjuncts as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var a end metavar imply not0 metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var b end metavar modus ponens metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var a end metavar imply not0 metavar var b end metavar infer not0 metavar var b end metavar conclude metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar cut metavar var a end metavar infer metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar cut prop lemma add double neg modus ponens metavar var b end metavar conclude not0 not0 metavar var b end metavar cut prop lemma mt modus ponens metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar modus ponens not0 not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Udskilning af anden konjunkt}
Tautologien " [ math prop lemma second conjunct end math ] " lader os udskille den anden konjunkt fra " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ". Jeg vil ikke kommentere beviset:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma second conjunct as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma second conjunct as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar infer prop lemma weakening modus ponens not0 metavar var b end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar infer metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar cut not0 metavar var a end metavar imply not0 metavar var b end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut prop lemma negative mt modus ponens not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Udskilning af f\o{}rste konjunkt}
For at udskille " [ math metavar var a end metavar end math ] " fra " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] " viser vi f\o{}rst, at " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] " er kommutativ. Jeg vil ikke kommentere beviset:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma and commutativity as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma and commutativity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply not0 metavar var a end metavar infer metavar var a end metavar infer prop lemma add double neg modus ponens metavar var a end metavar conclude not0 not0 metavar var a end metavar cut prop lemma mt modus ponens metavar var b end metavar imply not0 metavar var a end metavar modus ponens not0 not0 metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply not0 metavar var a end metavar infer metavar var a end metavar infer not0 metavar var b end metavar conclude metavar var b end metavar imply not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar cut not0 metavar var a end metavar imply not0 metavar var b end metavar infer 1rule repetition conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut prop lemma mt modus ponens metavar var b end metavar imply not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule repetition modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item Nu er det let at udskille den f\o{}rste konjunkt fra " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ": F\o{}rst vender vi konjunktionen om til " [ bracket not0 metavar var b end metavar imply not0 metavar var a end metavar end bracket ] " ved hj\ae{}lp af " [ math prop lemma and commutativity end math ] ", og s\aa{} udskiller vi " [ math metavar var a end metavar end math ] " ved hj\ae{}lp af " [ math prop lemma second conjunct end math ] ":
\item " [ math define statement of prop lemma first conjunct as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma first conjunct as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer prop lemma and commutativity modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut prop lemma second conjunct modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Dobbeltimplikation}
I dette underafsnit viser vi tre enkle resultater vedr.\ dobbeltimplikation.
\subsubsection{Brug sammen med modus ponens}
De f\o{}lgende to tautologier g\o{}r det let at bruge anvende slutningsreglen " [ math 1rule mp end math ] " p\aa{} dobbeltimplikationer. Beviserne er enkle og kr\ae{}ver ingen kommentarer:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma iff first as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma iff first as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar infer prop lemma second conjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude metavar var b end metavar imply metavar var a end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item" [ math define statement of prop lemma iff second as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var a end metavar infer metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma iff second as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var a end metavar infer prop lemma first conjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude metavar var a end metavar imply metavar var b end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Kommutativitet}
Lemmaet " [ math prop lemma iff commutativity end math ] " f\o{}lger direkte af, at operatoren " [ bracket not0 var x imply not0 var y end bracket ] "
er kommutativ:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma iff commutativity as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma iff commutativity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar cut prop lemma and commutativity modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Disjunktion}
Dette underafsnit indeholder tre lemmaer vedr.\ disjunktion, som vi fordeler p\aa{} to under-underafsnit.
\subsubsection{Sv\ae{}kkelse}
Givet en p\aa{}stand " [ math metavar var b end metavar end math ] " vil vi gerne udlede de svagere p\aa{}stande " [ bracket not0 metavar var a end metavar imply metavar var b end metavar end bracket ] " og " [ bracket not0 metavar var b end metavar imply metavar var a end metavar end bracket ] ". Den f\o{}rste slutning varetages af lemmaet " [ math prop lemma weaken or first end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma weaken or first as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item Beviset best\aa{}r af en simpel anvendelse af " [ math prop lemma weakening end math ] ":
\item " [ math define proof of prop lemma weaken or first as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer prop lemma weakening modus ponens metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\item Slutningen fra " [ math metavar var a end metavar end math ] " til " [ bracket not0 metavar var a end metavar imply metavar var b end metavar end bracket ] " varetages af lemmaet " [ math prop lemma weaken or second end math ] ":
\item " [ math define statement of prop lemma weaken or second as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item Kernen i beviset for " [ math prop lemma weaken or second end math ] " er en anvendelse af " [ math prop lemma from contradiction end math ] ":
\item " [ math define proof of prop lemma weaken or second as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer prop lemma from contradiction modus ponens metavar var a end metavar modus ponens not0 metavar var a end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Slutning ud fra disjunktion}
Lemmaet " [ math prop lemma from disjuncts end math ] " lader os drage slutninger ud fra en disjunktion:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma from disjuncts as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var c end metavar end define end math ] "
\item Om beviset vil jeg kun sige, at det er en ret elegant \o{}velse i bevisteknik:
\item " [ math define proof of prop lemma from disjuncts as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var c end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma contrapositive modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply not0 not0 metavar var a end metavar cut prop lemma technicality modus ponens metavar var a end metavar imply metavar var c end metavar conclude not0 not0 metavar var a end metavar imply metavar var c end metavar cut prop lemma imply transitivity modus ponens not0 metavar var b end metavar imply not0 not0 metavar var a end metavar modus ponens not0 not0 metavar var a end metavar imply metavar var c end metavar conclude not0 metavar var b end metavar imply metavar var c end metavar cut prop lemma contrapositive modus ponens not0 metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var c end metavar imply not0 not0 metavar var b end metavar cut prop lemma contrapositive modus ponens metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var c end metavar imply not0 metavar var b end metavar cut 1rule ad absurdum modus ponens not0 metavar var c end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var c end metavar imply not0 not0 metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\newpage
\section{Regellemaer} \label{sec:regel}
Alle aksiomerne i afsnit \ref{sec:axiomzf} er formuleret som dobbeltimplikationer. Dette er gjort for at holde antallet af aksiomer nede p\aa{} et minimum; men n\aa{}r aksiomerne skal bruges i beviser, er det mere bekvemt at have regler af formen " [ bracket metavar var a end metavar infer metavar var b end metavar end bracket ] " til r\aa{}dighed. Heldigvis kan vi altid bevise
" [ bracket metavar var a end metavar infer metavar var b end metavar end bracket ] ",
hvis vi i forvejen har et bevis for " [ bracket metavar var a end metavar imply metavar var b end metavar end bracket ] " --- der kr\ae{}ves blot en rutinem\ae{}ssig anvendelse af " [ math 1rule mp end math ] ".
I dette afsnit viser vi regellemmaer svarende til aksiomerne " [ math axiom separation definition end math ] ", " [ math axiom pair definition end math ] ", " [ math axiom union definition end math ] ", " [ math axiom power definition end math ] " og " [ math axiom extensionality end math ] ".
\subsection{Par} \label{sec:parlemma}
I tilf\ae{}ldet ``" [ math axiom pair definition end math ] "'' forl\o{}ber b\aa{}de definition og bevis af regellemmaerne s\aa{} enkelt, som det er muligt. F\o{}rst formulerer vi den ene halvdel af aksiomets dobbelt-implikation som et lemma:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of lemma pair2formula as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair infer not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar end define end math ] "
\item Derefter beviser vi dette lemma ved at anvende " [ math prop lemma iff second end math ] " p\aa{} aksiomet og pr\ae{}missen:
\item " [ math define proof of lemma pair2formula as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair infer axiom pair definition conclude not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair cut prop lemma iff second modus ponens not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair modus ponens metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair conclude not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\item Til sidst gentager vi hele processen mht.\ den anden halvdel af dobbeltimplikationen. Den eneste reelle forskel er, at vi bruger " [ math prop lemma iff first end math ] " i beviset (i stedet for " [ math prop lemma iff second end math ] "):
\item " [ math define statement of lemma formula2pair as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end define end math ] "
\item " [ math define proof of lemma formula2pair as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar infer axiom pair definition conclude not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair cut prop lemma iff first modus ponens not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair modus ponens not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar conclude metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Foreningsm\ae{}ngde}
I tilf\ae{}ldet ``" [ math axiom union definition end math ] "'' bruger vi stort set den samme fremgangsm\aa{}de som i afsnit \ref{sec:parlemma}. F\o{}rst h\aa{}ndterer vi den ene halvdel af dobbeltimplikationen:
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\item " [ math define statement of lemma union2formula as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in union metavar var x end metavar end union infer not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar end define end math ] "
\item " [ math define proof of lemma union2formula as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in union metavar var x end metavar end union infer axiom union definition conclude not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union cut prop lemma iff second modus ponens not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union modus ponens metavar var s end metavar zermelo in union metavar var x end metavar end union conclude not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\item S\aa{} er turen kommet til den anden halvdel af dobbeltimplikationen:
\item " [ math define statement of lemma formula2union as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in existential var var j end var infer existential var var j end var zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in union metavar var x end metavar end union end define end math ] "
\item Der er det lille raffinement, at vi i formuleringen af " [ math lemma formula2union end math ] " har delt formlen " [ bracket not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar end bracket ] " op i sine konjunkter. Beviset for " [ math lemma formula2union end math ] " kr\ae{}ver derfor en ekstra anvendelse af " [ math prop lemma join conjuncts end math ] ":
\item " [ math define proof of lemma formula2union as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in existential var var j end var infer existential var var j end var zermelo in metavar var x end metavar infer prop lemma join conjuncts modus ponens metavar var s end metavar zermelo in existential var var j end var modus ponens existential var var j end var zermelo in metavar var x end metavar conclude not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar cut axiom union definition conclude not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union cut prop lemma iff first modus ponens not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union modus ponens not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in union metavar var x end metavar end union end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Potensm\ae{}ngde; sidebetingelser} \label{sec:potside}
Tilf\ae{}ldet ``" [ math axiom power definition end math ] "'' er mere kompliceret end de hidtidige tilf\ae{}lde. Det ene af de to regellemmaer er nemt nok:
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\item " [ math define statement of lemma subset in power set as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in power metavar var x end metavar end power end define end math ] "
\item " [ math define proof of lemma subset in power set as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar infer 1rule gen modus ponens object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar cut axiom power definition conclude not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power cut prop lemma iff first modus ponens not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in power metavar var x end metavar end power end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent Det andet regellemma er ogs\aa{} nemt, hvis vi n\o{}jes med at vise:
" [ bracket all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in power metavar var x end metavar end power infer for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end bracket ] ".
\noindent For at komme videre med konklusionen " [ bracket for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end bracket ] "
m\aa{} vi imidlertid slippe af med objektkvantoren; og dette kan vi lige s\aa{} godt g\o{}re med det samme. Derfor bliver m\aa{}let i stedet at vise:
" [ bracket all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in power metavar var x end metavar end power infer object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end bracket ] ", hvor " [ math object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end math ] " som n\ae{}vnt i afsnit \ref{sec:subset} er en forkortelse for " [ bracket object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end bracket ] ".
For at n\aa{} dette m\aa{}l m\aa{} vi begynde med et hj\ae{}lpelemma, der ser s\aa{}ledes ud:
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\item " [ math define statement of lemma power set is subset0 as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar end define end math ] "
\end{list}
\noindent De to sidebetingelser " [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote end bracket ] " og " [ bracket quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote end bracket ] " er her n\o{}dvendige pga. antecedenten\footnote{Jeg kalder " [ math var x end math ] " i " [ bracket var x imply var y end bracket ] " for ``antecedenten''; og jeg kalder " [ math var y end math ] " for ``konsekvensen''.} " [ bracket for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] ". Hvis vi i denne formel instantierer " [ math metavar var x end metavar end math ] " eller " [ math metavar var y end metavar end math ] " til en term, der indeholder frie forekomster af " [ math object var var s end var end math ] ", s\aa{} \ae{}ndres formlens mening. F.eks.\ er formlen " [ bracket for all objects object var var s end var indeed object var var s end var zermelo in object var var s end var imply object var var s end var zermelo in object var var y end var end bracket ] " n\o{}dvendigvis sand, fordi ingen m\ae{}ngder i " [ math system zf end math ] " er medlem af sig selv; men den oprindelige formel " [ bracket for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] " er ikke n\o{}dvendigvis sand. Konstruktionen " [ bracket quote var x end quote avoid zero quote var y end quote end bracket ] " (der er defineret i appendikset til \cite{kn:check}) udsiger netop, at objektvariablen " [ math var x end math ] " ikke forekommer frit i termen " [ math var y end math ] "\footnote{Vi siger at ``" [ math var x end math ] " undg\aa{}r " [ math var y end math ] "'' (eller at ``" [ math var y end math ] " undg\aa{}r " [ math var x end math ] "'').}. Vi kommer til at se mange flere eksempler p\aa{} sidebetingelser af formen " [ bracket quote var x end quote avoid zero quote var y end quote end bracket ] ": De er n\o{}dvendige, n\aa{}r metavariablen " [ math var y end math ] " optr\ae{}der i en kontekst, der er kvantificeret med objektvariablen " [ math var x end math ] ".
Indholdet af selve lemmaet er, at vi kan fjerne objektkvantoren fra
\\ \mbox{" [ bracket for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] "}, og endda erstatte objektvariablen " [ math object var var s end var end math ] " med metavariablen " [ math metavar var s end metavar end math ] ". Her er beviset:
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\item " [ math define proof of lemma power set is subset0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar infer 1rule repetition modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar conclude object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar infer object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent Beviset illustrerer styrken af slutningsreglen " [ math 1rule deduction end math ] ": I dette tilf\ae{}lde omdanner den en inferens (nemlig " [ bracket all metavar var x end metavar indeed all metavar var y end metavar indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar infer object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] ") til en implikation (nemlig " [ bracket object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] "), s\ae{}tter en alkvantor p\aa{} antecedenten (s\aa{} vi f\aa{}r " [ bracket for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] "), og erstatter den frie forekomst af ``" [ math object var var s end var end math ] "'' med ``" [ math metavar var s end metavar end math ] "'' --- alt sammen p\aa{} \'{e}n gang. Dog skal de to sidebetingelser med, for at reglen kan anvendes.
Formuleringen af det andet regellemma bliver nu som f\o{}lger:
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\item " [ math define statement of lemma power set is subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in power metavar var x end metavar end power infer object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end define end math ] "
\end{list}
\noindent Da " [ math lemma power set is subset end math ] " benytter sig af " [ math lemma power set is subset0 end math ] ", bliver vi n\o{}dt til at lade " [ math lemma power set is subset end math ] " ``arve'' de to sidebetingelser fra
%\\ \mbox{" [ math lemma power set is subset0 end math ] "}
dette lemma. Vi kan ikke undg\aa{} sidebetingelserne ved at instantiere de to metavariable " [ math metavar var x end metavar end math ] " og " [ math metavar var y end metavar end math ] " fra " [ math lemma power set is subset0 end math ] " til nogle andre metavariable; for der er jo ingen garanti for, at disse metavariable vil undg\aa{} " [ math object var var s end var end math ] ", n\aa{}r de engang bliver instantierede. S\aa{} l\ae{}nge man arbejder med metavariable, er det alts\aa{} meget sv\ae{}rt at slippe af med denne slags sidebetingelser, n\aa{}r de f\o{}rst er blevet indf\o{}rt i en bevisk\ae{}de.
Her er beviset for " [ math lemma power set is subset end math ] ":
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\item " [ math define proof of lemma power set is subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in power metavar var x end metavar end power infer axiom power definition conclude not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power cut prop lemma iff second modus ponens not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power modus ponens metavar var s end metavar zermelo in power metavar var x end metavar end power conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma power set is subset0 modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar cut 1rule mp modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar cut 1rule repetition modus ponens object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\item I linie 7 benytter vi konstruktionen " [ bracket var x modus probans var y end bracket ] " til at fort\ae{}lle bevischeckeren, at de to sidebetingelser er opfyldt, og at vi derfor kan anvende " [ math lemma power set is subset0 end math ] ".
\end{list}
\subsection{Potensm\ae{}ngde, variant}
Som n\ae{}vnt er det sv\ae{}rt at slippe af med sidebetingelser, men af og til kan det lade sig g\o{}re.
N\aa{}r vi i afsnit \ref{sec:union2} g\o{}r brug af lemmaerne " [ math lemma power set is subset0 end math ] " og " [ math lemma power set is subset end math ] ", vil metavariablen " [ math metavar var x end metavar end math ] " f.eks.\ blive instantieret til termer, der undg\aa{}r " [ math object var var s end var end math ] ". Hermed bliver sidebetingelsen " [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote end bracket ] " automatisk opfyldt, mens vi stadigv\ae{}k m\aa{} medregne " [ bracket quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote end bracket ] " eksplicit. Imidlertid er bevischeckeren s\aa{}ledes indrettet, at de automatisk opfyldte sidebetingelser skal st\aa{} til sidst, n\aa{}r man bruger et lemma, der indeholder sidebetingelser. Derfor viser vi nu nogle varianter af " [ math lemma power set is subset end math ] " og " [ math lemma power set is subset0 end math ] ", hvor der er byttet om p\aa{} " [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote end bracket ] " og " [ bracket quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote end bracket ] ":
\begin{list}{}{
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\item " [ math define statement of lemma power set is subset0-switch as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar end define end math ] "
\item " [ math define proof of lemma power set is subset0-switch as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse lemma power set is subset0 modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma power set is subset-switch as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var s end metavar zermelo in power metavar var x end metavar end power infer object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end define end math ] "
\item " [ math define proof of lemma power set is subset-switch as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var s end metavar zermelo in power metavar var x end metavar end power infer lemma power set is subset modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus ponens metavar var s end metavar zermelo in power metavar var x end metavar end power conclude object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent Vi kunne naturligvis have sparet noget plads ved at vise disse varianter til at begynde med. Jeg har ikke gjort dette, fordi vi nu har f\aa{}et illustreret en af ulemperne ved at arbejde med sidebetingelser.
Jeg vil ikke kommentere f\ae{}nomenet ``sidebetingelser af formen " [ bracket quote var x end quote avoid zero quote var y end quote end bracket ] "'' yderligere --- indtil afsnit \ref{sec:binaerforen}, hvor antallet af sidebetingelser bliver s\aa{} stort, at det ikke kan ignoreres.
\subsection{Separation} \label{sec:seplemma}
Tilf\ae{}ldet ``" [ math axiom separation definition end math ] "'' er n\ae{}sten lige s\aa{} enkelt som tilf\ae{}ldet ``" [ math axiom pair definition end math ] "'' fra afsnit \ref{sec:parlemma}. Vi skal blot huske at overf\o{}re aksiomets sidebetingelse til de to regellemmaer:
\begin{list}{}{
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\item " [ math define statement of lemma separation2formula as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub endorse metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set infer not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar end define end math ] "
\item " [ math define proof of lemma separation2formula as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub endorse metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set infer axiom separation definition modus probans metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub conclude not0 metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set cut prop lemma first conjunct modus ponens not0 metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set conclude metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar cut 1rule mp modus ponens metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar modus ponens metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set conclude not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma formula2separation as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub endorse metavar var y end metavar zermelo in metavar var x end metavar infer metavar var b end metavar infer metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set end define end math ] "
\item " [ math define proof of lemma formula2separation as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub endorse metavar var y end metavar zermelo in metavar var x end metavar infer metavar var b end metavar infer prop lemma join conjuncts modus ponens metavar var y end metavar zermelo in metavar var x end metavar modus ponens metavar var b end metavar conclude not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar cut axiom separation definition modus probans metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var y end metavar end quote end sub conclude not0 metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set cut prop lemma second conjunct modus ponens not0 metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set conclude not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set cut 1rule mp modus ponens not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set modus ponens not0 metavar var y end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar conclude metavar var y end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Ekstentionalitet}
Tilf\ae{}ldet ``" [ math axiom extensionality end math ] "'' er lidt mere kompliceret end det almindelige tilf\ae{}lde, fordi vi skal forholde os til objektkvantoren i meta-formlen " [ bracket not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ". Derfor er der to underafsnit; lemmaerne i underafsnit \ref{sec:tillighed} er afledt af implikationen " [ bracket for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ", mens lemmaerne i underafsnit \ref{sec:fralighed} er afledt af den omvendte implikation.
\subsubsection{Tilstr\ae{}kkelig betingelse for lighed}
\label{sec:tillighed}
To m\ae{}ngder er lig hinanden, hvis de er hinandens delm\ae{}ngder:
\begin{list}{}{
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\item " [ math define statement of lemma set equality suff condition as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar infer object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is metavar var y end metavar end define end math ] "
\item Beviset for " [ math lemma set equality suff condition end math ] " er ret enkelt. Vi skal blot huske at s\ae{}tte en objektkvantor p\aa{} pr\ae{}misserne, hvilket g\o{}res med slutningsreglen " [ math 1rule gen end math ] ":
\item " [ math define proof of lemma set equality suff condition as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar infer object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar infer prop lemma join conjuncts modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar modus ponens object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut 1rule gen modus ponens not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut axiom extensionality conclude not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut 1rule mp modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude metavar var x end metavar zermelo is metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\item I afsnit \ref{sec:parlig} f\aa{}r vi brug for en version af dette lemma, hvor objektvariablen " [ math object var var s end var end math ] " er erstattet af en anden objektvariabel (jeg har valgt " [ math object var var t end var end math ] "). For at vise denne alternative version viser vi f\o{}rst lemmaet " [ math lemma set equality suff condition(t)0 end math ] ", som indkapsler en anvendelse af deduktionsreglen:
\item " [ math define statement of lemma set equality suff condition(t)0 as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end define end math ] "
\item " [ math define proof of lemma set equality suff condition(t)0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar infer 1rule repetition modus ponens object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar conclude object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar infer object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar conclude quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\item Hovedlemmaet hedder " [ math lemma set equality suff condition(t) end math ] ":
\item " [ math define statement of lemma set equality suff condition(t) as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote endorse object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar infer object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is metavar var y end metavar end define end math ] "
\item I beviset herfor kombinerer vi " [ math 1rule gen end math ] " med hj\ae{}lpelemmaet til at omdanne pr\ae{}missen " [ bracket object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar end bracket ] " til " [ bracket object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] " (linie 6--8). Vi gentager denne procedure mht.\ pr\ae{}missen " [ bracket object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar end bracket ] " (linie 9-11). Vi kan da slutte af med en anvendelse af det oprindelige lemma " [ math lemma set equality suff condition end math ] " (linie 12):
\item " [ math define proof of lemma set equality suff condition(t) as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote endorse object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar infer object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar infer 1rule gen modus ponens object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar conclude for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar cut lemma set equality suff condition(t)0 modus probans quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote conclude for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar cut 1rule mp modus ponens for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar modus ponens for all objects object var var t end var indeed object var var t end var zermelo in metavar var x end metavar imply object var var t end var zermelo in metavar var y end metavar conclude object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar cut 1rule gen modus ponens object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar conclude for all objects object var var t end var indeed object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar cut lemma set equality suff condition(t)0 modus probans quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote conclude for all objects object var var t end var indeed object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut 1rule mp modus ponens for all objects object var var t end var indeed object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar modus ponens for all objects object var var t end var indeed object var var t end var zermelo in metavar var y end metavar imply object var var t end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar modus ponens object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude metavar var x end metavar zermelo is metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
Arbejdsfordelingen mellem " [ math lemma set equality suff condition(t)0 end math ] " og " [ math lemma set equality suff condition(t) end math ] " er temmelig uj\ae{}vn; det meste af arbejdet foreg\aa{}r i beviset for " [ math lemma set equality suff condition(t) end math ] ". At der er to lemmaer i stedet for \'{e}t, har en teknisk forklaring: Logiwebs bevischecker kan ikke arbejde med konklusioner af formen " [ bracket var x endorse var y end bracket ] ". Vi kan derfor kun referere til konklusionen " [ bracket quote object var var t end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var y end metavar end quote endorse cdots end bracket ] " ved at lade den v\ae{}re konklusionen p\aa{} et bevis; og derfor m\aa{} vi stoppe op, n\aa{}r vi n\aa{}r til linie 7 i " [ math lemma set equality suff condition(t)0 end math ] ". Vi vil flere gange senere komme ud for lemmaer, hvis eksistens har denne tekniske forklaring. Jeg vil bruge ordet ``lemmastump'' om disse lemmaer.
\subsubsection{N\o{}dvendig betingelse for lighed} \label{sec:fralighed}
Vi begynder dette underafsnit med et hj\ae{}lpelemma, som vi bruger til at fjerne objektkvantoren fra " [ math axiom extensionality end math ] ":
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\item " [ math define statement of lemma set equality skip quantifier as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar end define end math ] "
\item Beviset for " [ math lemma set equality skip quantifier end math ] " er en simpel anvendelse af deduktionsreglen:
\item " [ math define proof of lemma set equality skip quantifier as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar infer 1rule repetition modus ponens not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar infer not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\item Hovelemmaet i dette underafsnit udsiger, at hvis " [ math metavar var x end metavar end math ] " er lig med " [ math metavar var y end metavar end math ] ", s\aa{} er " [ math metavar var x end metavar end math ] " en delm\ae{}ngde af " [ math metavar var y end metavar end math ] "\footnote{Naturligvis kan man ogs\aa{} vise, at hvis " [ math metavar var x end metavar end math ] " er lig med " [ math metavar var y end metavar end math ] ", s\aa{} er " [ math metavar var y end metavar end math ] " en delm\ae{}ngde af " [ math metavar var x end metavar end math ] ". Det har imidlertid ikke v\ae{}ret n\o{}dvendigt at bruge dette lemma.}:
\item " [ math define statement of lemma set equality nec condition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in metavar var y end metavar end define end math ] "
\item Beviset er en enkel anvendelse af " [ math axiom extensionality end math ] " og " [ math lemma set equality skip quantifier end math ] ":
\item " [ math define proof of lemma set equality nec condition as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer axiom extensionality conclude not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut prop lemma iff second modus ponens not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality skip quantifier modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar cut 1rule mp modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar cut prop lemma iff second modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\section{Definitionslemmaer} \label{sec:def}
Regellemmaerne fra afsnit \ref{sec:regel} blev afledt fra aksiomer. Lemmaerne i dette afsnit bliver afledt fra makrodefinitionerne i afsnit \ref{sec:eqrel}; heraf navnet ``definitionslemmaer''. Form\aa{}let med definitionslemmaerne er at muligg\o{}re effektiv anvendelse af makrodefinitionerne i beviser.
\subsection{Refleksiv relation} \label{sec:refldef}
Den f\o{}rste definition, vi tager under behandling, er definitionen
\\ \mbox{" [ bracket define-equal for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar comma for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end equal end bracket ] "}. Vi bruger her en lemmastump ved navn " [ math lemma reflexivity0 end math ] ", som lader os fjerne objektkvantoren fra denne definitions h\o{}jreside:
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\item " [ math define statement of lemma reflexivity0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item Vi viser " [ math lemma reflexivity0 end math ] " ved hj\ae{}lp af deduktionsreglen:
\item " [ math define proof of lemma reflexivity0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\item I hovedlemmaet " [ math lemma reflexivity end math ] " omdanner vi lemmastumpens implikationer til inferenser:
\item " [ math define statement of lemma reflexivity as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma reflexivity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer lemma reflexivity0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma mp2 modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent Vi skal straks se flere eksempler p\aa{} netop denne arbejdsdeling mellem lemmastump og hovedlemma.
\subsection{Symmetrisk relation} \label{sec:symreldef}
De to lemmaer i dette underafsnit er afledt fra definitionen af ``symmetrisk relation'' i afsnit \ref{sec:eqrel}. Fremgangsm\aa{}den er den samme som i afsnit \ref{sec:refldef}. Lemmaet " [ math lemma symmetry0 end math ] " fungerer som lemmastump i forhold til " [ math lemma symmetry end math ] ":
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\item " [ math define statement of lemma symmetry0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define statement of lemma symmetry as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var t end metavar zermelo in metavar var big set end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item Beviset for " [ math lemma symmetry0 end math ] " fungerer ligesom beviset for " [ math lemma reflexivity0 end math ] " fra afsnit \ref{sec:refldef} (dog skriver vi ikke definitionen af " [ math for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end math ] " ud):
\item " [ math define proof of lemma symmetry0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\item I beviset for " [ math lemma symmetry end math ] " skifter vi fra implikation til inferens:
\item " [ math define proof of lemma symmetry as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var t end metavar zermelo in metavar var big set end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar infer lemma symmetry0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma mp4 modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens metavar var t end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Transitiv relation} \label{sec:transdef}
De to lemmaer i dette underafsnit er afledt fra definitionen af ``transitiv relation'' i afsnit \ref{sec:eqrel}. Vi bruger n\o{}jagtig den samme fremgangsm\aa{}de som i afsnit \ref{sec:refldef} og \ref{sec:symreldef}: Lemmastumpen (" [ math lemma transitivity0 end math ] ") fjerner definitionens objektkvantor ved hj\ae{}lp af " [ math 1rule deduction end math ] ", og hovedlemmaet (" [ math lemma transitivity end math ] ") skifter fra implikation til inferens:
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\item " [ math define statement of lemma transitivity0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var u end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply metavar var u end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma transitivity0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var u end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var big set end metavar indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply metavar var u end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma transitivity as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var u end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var t end metavar zermelo in metavar var big set end metavar infer metavar var u end metavar zermelo in metavar var big set end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma transitivity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var t end metavar indeed all metavar var u end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var t end metavar zermelo in metavar var big set end metavar infer metavar var u end metavar zermelo in metavar var big set end metavar infer zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar infer lemma transitivity0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply metavar var u end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma mp5 modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var t end metavar zermelo in metavar var big set end metavar imply metavar var u end metavar zermelo in metavar var big set end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens metavar var t end metavar zermelo in metavar var big set end metavar modus ponens metavar var u end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var t end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar cut 1rule mp modus ponens zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var t end metavar comma metavar var t end metavar end pair comma zermelo pair metavar var t end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var u end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{\AE{}kvivalensrelation}
De tre lemmaer i dette underafsnit udsiger, at en \ae{}kvivalensrelation er hhv.\ refleksiv, symmetrisk og transitiv. Beviserne er simple; vi anvender et par tautologier p\aa{} definitionen af " [ math not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar end math ] " fra afsnit \ref{sec:eqrel}:
\begin{list}{}{
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\item " [ math define statement of lemma er is reflexive as system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma er is reflexive as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma er is symmetric as system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma er is symmetric as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma er is transitive as system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item " [ math define proof of lemma er is transitive as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer 1rule repetition modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\section{Basale s\ae{}tninger i m\ae{}ngdel\ae{}re} \label{sec:basal}
Lemmaerne i dette afsnit er en l\o{}s samling af m\ae{}ngdeteoretiske s\ae{}tninger, som bliver brugt i beviset for hovedresultatet i afsnit \ref{sec:hoved}. Lemmaerne er grupperet efter emne: Underafsnit \ref{sec:empty} omhandler den tomme m\ae{}ngde, underafsnit \ref{sec:lighed} handler om lighedsrelationen " [ bracket var x zermelo is var y end bracket ] ", og underafsnit \ref{sec:notequal} handler om negeret lighed (" [ math not0 var x zermelo is var y end math ] "). Figur 2 giver et overblik over lemmaerne.
\clearpage
% figur 2
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,22cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/diverse.eps}}
\caption[Bevisstrukturen for afsnit \ref{sec:basal}]{Bevisstrukturen for afsnit \ref{sec:basal}. Kun lemmaer fra dette afsnit er vist. Tallene i parentes angiver, hvor mange sidebetingelser de forskellige lemmaer indeholder. En pil fra lemma " [ math var x end math ] " til lemma " [ math var y end math ] " betyder, at " [ math var x end math ] " bruges i beviset for " [ math var y end math ] ". Tallene ved pilene angiver, hvor mange sidebetingelser " [ math var y end math ] " modtager fra " [ math var x end math ] ". Hvis der f.eks.\ st\aa{}r ``$2 \cdot \mathbf{4}$'', betyder det, at " [ math var x end math ] " bliver anvendt to gange i beviset for " [ math var y end math ] ", begge gange med 4 sidebetingelser.
Der er overlap mellem nogle af sidebetingelserne, og lemmaer kan generere deres egne sidebetingelser. Derfor er der ikke n\o{}dvendigvis overensstemmelse mellem tallene i parentes og tallene ved lemmaerne.}
\end{figure}
\clearpage
\subsection{Den tomme m\ae{}ngde} \label{sec:empty}
Dette underafsnit indeholder 3 lemmaer, der hver har f\aa{}et sit under-underafsnit.
\subsubsection{En delm\ae{}ngde af alle m\ae{}ngder}
Vi viser f\o{}rst, at den tomme m\ae{}ngde er en delm\ae{}ngde af alle m\ae{}ngder:
\begin{list}{}{
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\item " [ math define statement of lemma empty set is subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in zermelo empty set imply metavar var s end metavar zermelo in metavar var x end metavar end define end math ] "
\item Beviset for " [ math lemma empty set is subset end math ] " benytter sig af deduktionsreglen og tautologien \\ \mbox{" [ math prop lemma from contradiction end math ] "}:
\item " [ math define proof of lemma empty set is subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in zermelo empty set infer axiom empty set conclude not0 metavar var s end metavar zermelo in zermelo empty set cut prop lemma from contradiction modus ponens metavar var s end metavar zermelo in zermelo empty set modus ponens not0 metavar var s end metavar zermelo in zermelo empty set conclude metavar var s end metavar zermelo in metavar var x end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed metavar var s end metavar zermelo in zermelo empty set infer metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in zermelo empty set imply metavar var s end metavar zermelo in metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Der er kun \'{e}n tom m\ae{}ngde}
Det n\ae{}ste lemma beviser p\aa{}standen ``der er kun \'{e}n tom m\ae{}ngde''. Vi formulerer denne p\aa{}stand som f\o{}lger: ``Hvis en m\ae{}ngde " [ math metavar var x end metavar end math ] " ikke har nogen medlemmer, s\aa{}
er " [ math metavar var x end metavar end math ] " lig med " [ math zermelo empty set end math ] "'':
\begin{list}{}{
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\item " [ math define statement of lemma unique empty set as system zf infer all metavar var x end metavar indeed not0 object var var s end var zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is zermelo empty set end define end math ] "
\item Vi ved allerede fra lemmaet " [ math lemma empty set is subset end math ] ", at " [ math zermelo empty set end math ] " er en delm\ae{}ngde af " [ math metavar var x end metavar end math ] ". Alts\aa{} mangler vi blot at vise, at " [ math metavar var x end metavar end math ] " er en delm\ae{}ngde af " [ math zermelo empty set end math ] ". Dette klarer vi med hj\ae{}lpelemmaet " [ math lemma unique empty set0 end math ] ":
\item " [ math define statement of lemma unique empty set0 as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in zermelo empty set end define end math ] "
\item " [ math define proof of lemma unique empty set0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer prop lemma from contradiction modus ponens metavar var s end metavar zermelo in metavar var x end metavar modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in zermelo empty set cut all metavar var s end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in zermelo empty set conclude not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in zermelo empty set cut not0 metavar var s end metavar zermelo in metavar var x end metavar infer 1rule mp modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in zermelo empty set modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item Med " [ math lemma unique empty set0 end math ] " til r\aa{}dighed er det let at vise " [ math lemma unique empty set end math ] ":
\item " [ math define statement of lemma unique empty set as system zf infer all metavar var x end metavar indeed not0 object var var s end var zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma unique empty set as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed not0 object var var s end var zermelo in metavar var x end metavar infer lemma unique empty set0 modus ponens not0 object var var s end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in zermelo empty set cut lemma empty set is subset conclude object var var s end var zermelo in zermelo empty set imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in zermelo empty set modus ponens object var var s end var zermelo in zermelo empty set imply object var var s end var zermelo in metavar var x end metavar conclude metavar var x end metavar zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Lemmaet ``" [ math lemma member not empty end math ] "''}
\label{sec:membernotempty}
Som det sidste resultat i dette underafsnit viser vi, at en m\ae{}ngde, der har et medlem, ikke er lig med den tomme m\ae{}ngde:
\begin{list}{}{
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\item " [ math define statement of lemma member not empty as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer not0 metavar var x end metavar zermelo is zermelo empty set end define end math ] "
\item Beviset for " [ math lemma member not empty end math ] " kan gengives som f\o{}lger:
%
\begin{enumerate}
\item ``Antag at " [ math metavar var s end metavar end math ] " tilh\o{}rer " [ math metavar var x end metavar end math ] ". Hvis " [ math metavar var x end metavar end math ] " var lig med " [ math zermelo empty set end math ] ", s\aa{} ville " [ math metavar var s end metavar end math ] " ogs\aa{} tilh\o{}re " [ math zermelo empty set end math ] "''.
\item ``Men " [ math metavar var s end metavar end math ] " tilh\o{}rer ikke " [ math zermelo empty set end math ] ". Derfor er " [ math metavar var x end metavar end math ] " ikke lig med " [ math zermelo empty set end math ] "''.
\end{enumerate}
\item Vi viser punkt 1 som lemmastumpen " [ math lemma member not empty0 end math ] "; herefter gennemf\o{}rer vi punkt 2 i selve beviset for " [ math lemma member not empty end math ] ":
\item " [ math define statement of lemma member not empty0 as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set end define end math ] "
\item " [ math define proof of lemma member not empty0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is zermelo empty set infer lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus ponens metavar var x end metavar zermelo is zermelo empty set modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in zermelo empty set cut all metavar var s end metavar indeed all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is zermelo empty set infer metavar var s end metavar zermelo in zermelo empty set conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma member not empty as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer not0 metavar var x end metavar zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma member not empty as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer lemma member not empty0 modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote conclude metavar var s end metavar zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set cut 1rule mp modus ponens metavar var s end metavar zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set cut axiom empty set conclude not0 metavar var s end metavar zermelo in zermelo empty set cut prop lemma mt modus ponens metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set modus ponens not0 metavar var s end metavar zermelo in zermelo empty set conclude not0 metavar var x end metavar zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{ " [ bracket var x zermelo is var y end bracket ] " er en \ae{}kvivalensrelation } \label{sec:lighed}
M\aa{}let med dette underafsnit er at vise, at " [ bracket var x zermelo is var y end bracket ] " er en refleksiv, symmetrisk og transitiv relation.
At " [ bracket var x zermelo is var y end bracket ] " er refleksiv, f\o{}lger af, at " [ bracket var x imply var y end bracket ] " er refleksiv:
\begin{list}{}{
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\item " [ math define statement of lemma =reflexivity as system zf infer all metavar var s end metavar indeed metavar var s end metavar zermelo is metavar var s end metavar end define end math ] "
\item " [ math define proof of lemma =reflexivity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed prop lemma auto imply conclude object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var s end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var s end metavar modus ponens object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var s end metavar conclude metavar var s end metavar zermelo is metavar var s end metavar end quote state proof state cache var c end expand end define end math ] "
\item \indent \verb| | Symmetri-lemmaet for " [ bracket var x zermelo is var y end bracket ] " ser s\aa{}ledes ud:
\item " [ math define statement of lemma =symmetry as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var x end metavar end define end math ] "
\item Vi beviser " [ math lemma =symmetry end math ] " ved at reducere termen " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] " til de to implikationer " [ bracket object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar end bracket ] " og " [ bracket object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar end bracket ] ". Herefter s\ae{}tter vi de to implikationer sammen i omvendt r\ae{}kkef\o{}lge, hvilket giver os " [ bracket metavar var y end metavar zermelo is metavar var x end metavar end bracket ] ":
\item " [ math define proof of lemma =symmetry as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer axiom extensionality conclude not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut prop lemma iff second modus ponens not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality skip quantifier modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote conclude for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut 1rule mp modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar modus ponens for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut prop lemma first conjunct modus ponens not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar cut prop lemma second conjunct modus ponens not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar conclude object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar conclude metavar var y end metavar zermelo is metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
\item \indent \verb| | Transitivitets-lemmaet for " [ bracket var x zermelo is var y end bracket ] " ser s\aa{}ledes ud:
\item " [ math define statement of lemma =transitivity as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var z end metavar infer metavar var x end metavar zermelo is metavar var z end metavar end define end math ] "
\item Vi viser f\o{}rst den svagere p\aa{}stand, at " [ math metavar var x end metavar end math ] " er en delm\ae{}ngde af " [ math metavar var z end metavar end math ] ":
\item " [ math define statement of lemma =transitivity0 as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var z end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var z end metavar end define end math ] "
\item Bevisets kerne best\aa{}r af to anvendelser af " [ math lemma set equality nec condition end math ] " (linie 8--9):
\item " [ math define proof of lemma =transitivity0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var z end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in metavar var y end metavar cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote modus ponens metavar var y end metavar zermelo is metavar var z end metavar modus ponens metavar var s end metavar zermelo in metavar var y end metavar conclude metavar var s end metavar zermelo in metavar var z end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var z end metavar infer metavar var s end metavar zermelo in metavar var x end metavar infer metavar var s end metavar zermelo in metavar var z end metavar conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var z end metavar imply metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
\item Vi bruger nu " [ math lemma =transitivity0 end math ] " til at vise, at " [ math metavar var x end metavar end math ] " er en delm\ae{}ngde af " [ math metavar var z end metavar end math ] " (linie 5--8), og vice versa (line 9--12):
\item " [ math define statement of lemma =transitivity as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var z end metavar infer metavar var x end metavar zermelo is metavar var z end metavar end define end math ] "
\item " [ math define proof of lemma =transitivity as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var y end metavar zermelo is metavar var z end metavar infer lemma =transitivity0 modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote conclude metavar var x end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar cut prop lemma mp2 modus ponens metavar var x end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var y end metavar zermelo is metavar var z end metavar conclude object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var y end metavar zermelo is metavar var x end metavar cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote modus ponens metavar var y end metavar zermelo is metavar var z end metavar conclude metavar var z end metavar zermelo is metavar var y end metavar cut lemma =transitivity0 modus probans quote object var var s end var end quote avoid zero quote metavar var z end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote conclude metavar var z end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar cut prop lemma mp2 modus ponens metavar var z end metavar zermelo is metavar var y end metavar imply metavar var y end metavar zermelo is metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar modus ponens metavar var z end metavar zermelo is metavar var y end metavar modus ponens metavar var y end metavar zermelo is metavar var x end metavar conclude object var var s end var zermelo in metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var z end metavar modus ponens object var var s end var zermelo in metavar var z end metavar imply object var var s end var zermelo in metavar var x end metavar conclude metavar var x end metavar zermelo is metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Negeret lighed} \label{sec:notequal}
I dette sidste underafsnit viser vi, at lige st\o{}rrelser kan erstatte hinanden i en negeret lighed:
\begin{list}{}{
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\item " [ math define statement of lemma transfer ~is as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse not0 metavar var x end metavar zermelo is metavar var y end metavar infer metavar var x end metavar zermelo is metavar var v end metavar infer metavar var y end metavar zermelo is metavar var w end metavar infer not0 metavar var v end metavar zermelo is metavar var w end metavar end define end math ] "
\item Beviset for " [ math lemma transfer ~is end math ] " kan gengives som f\o{}lger:
%
\begin{enumerate}
\item ``Antag at " [ math metavar var v end metavar end math ] " er lig med " [ math metavar var w end metavar end math ] ". Ud fra pr\ae{}misserne " [ bracket metavar var x end metavar zermelo is metavar var v end metavar end bracket ] " og " [ bracket metavar var y end metavar zermelo is metavar var w end metavar end bracket ] " f\aa{}r vi da " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ".''
\item ``Men vi har antaget " [ bracket not0 metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ". Derfor kan " [ math metavar var v end metavar end math ] " ikke v\ae{}re lig med " [ math metavar var w end metavar end math ] ".''
\end{enumerate}
\item Vi viser punkt 1 som lemmastumpen " [ math lemma transfer ~is0 end math ] "; punkt 2 klares i selve beviset for " [ math lemma transfer ~is end math ] ":
\item " [ math define statement of lemma transfer ~is0 as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var v end metavar imply metavar var y end metavar zermelo is metavar var w end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end define end math ] "
\item " [ math define proof of lemma transfer ~is0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var v end metavar infer metavar var y end metavar zermelo is metavar var w end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer lemma =transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var v end metavar modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude metavar var x end metavar zermelo is metavar var w end metavar cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus ponens metavar var y end metavar zermelo is metavar var w end metavar conclude metavar var w end metavar zermelo is metavar var y end metavar cut lemma =transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var w end metavar modus ponens metavar var w end metavar zermelo is metavar var y end metavar conclude metavar var x end metavar zermelo is metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var v end metavar infer metavar var y end metavar zermelo is metavar var w end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer metavar var x end metavar zermelo is metavar var y end metavar conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var v end metavar imply metavar var y end metavar zermelo is metavar var w end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
\newpage
\item " [ math define statement of lemma transfer ~is as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse not0 metavar var x end metavar zermelo is metavar var y end metavar infer metavar var x end metavar zermelo is metavar var v end metavar infer metavar var y end metavar zermelo is metavar var w end metavar infer not0 metavar var v end metavar zermelo is metavar var w end metavar end define end math ] "
\item " [ math define proof of lemma transfer ~is as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse not0 metavar var x end metavar zermelo is metavar var y end metavar infer metavar var x end metavar zermelo is metavar var v end metavar infer metavar var y end metavar zermelo is metavar var w end metavar infer lemma transfer ~is0 modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote conclude metavar var x end metavar zermelo is metavar var v end metavar imply metavar var y end metavar zermelo is metavar var w end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut prop lemma mp2 modus ponens metavar var x end metavar zermelo is metavar var v end metavar imply metavar var y end metavar zermelo is metavar var w end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var x end metavar zermelo is metavar var v end metavar modus ponens metavar var y end metavar zermelo is metavar var w end metavar conclude metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar cut prop lemma mt modus ponens metavar var v end metavar zermelo is metavar var w end metavar imply metavar var x end metavar zermelo is metavar var y end metavar modus ponens not0 metavar var x end metavar zermelo is metavar var y end metavar conclude not0 metavar var v end metavar zermelo is metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\section{Lighedslemmaer} \label{sec:lighedslemmaer}
Ved et ``un\ae{}rt lighedslemma'' vil jeg forst\aa{} et lemma af f\o{}lgende form: ``Hvis der g\ae{}lder " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ", s\aa{} g\ae{}lder " [ bracket op metavar var x end metavar end op zermelo is op metavar var y end metavar end op end bracket ] "'' --- hvor `` " [ bracket op metavar var x end metavar end op end bracket ] "'' st\aa{}r for en un\ae{}r syntaktisk konstruktion som f.eks.\ " [ bracket union var x end union end bracket ] " eller " [ bracket zermelo pair var x comma var x end pair end bracket ] ".
Tilsvarende kan vi definere et ``bin\ae{}rt lighedslemma'' som et lemma med indholdet: ``Hvis der g\ae{}lder " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] " og " [ bracket metavar var v end metavar zermelo is metavar var w end metavar end bracket ] ", s\aa{} g\ae{}lder
" [ bracket op2 metavar var x end metavar comma metavar var v end metavar end op2 zermelo is op2 metavar var y end metavar comma metavar var w end metavar end op2 end bracket ] "'' --- her kan ``" [ bracket op2 var x comma var y end op2 end bracket ] "'' f.eks.\ st\aa{} for " [ bracket zermelo pair var x comma var y end pair end bracket ] " eller " [ bracket the set of ph in union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union such that not0 placeholder-var var c end var zermelo in var x imply not0 placeholder-var var c end var zermelo in var y end set end bracket ] ".
I dette afsnit beviser vi lighedslemmaerne for konstruktionerne
" [ bracket zermelo pair var x comma var y end pair end bracket ] ", " [ bracket zermelo pair var x comma var x end pair end bracket ] ", " [ bracket union var x end union end bracket ] ", " [ bracket union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union end bracket ] ", " [ bracket the set of ph in var x such that var f end set end bracket ] " og " [ bracket the set of ph in union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union such that not0 placeholder-var var c end var zermelo in var x imply not0 placeholder-var var c end var zermelo in var y end set end bracket ] ". Figur 3 giver et overblik over afsnittets lemmaer. Hele arbejdet i dette afsnit munder ud i \'{e}n eneste anvendelse af det sidste lighedslemma (``" [ math lemma same intersection end math ] "'') i beviset for hovedresultatet (i afsnit \ref{sec:alldisjoint}). Man kan alts\aa{} godt sige, at vi skyder gr\aa{}spurve med kanoner. Lighedslemmaer har dog ogs\aa{} interesse derved, at de kan bruges til at vise, at " [ math system zf end math ] " er en teori med lighed (jvf.\ afsnit 2.8 i \cite{kn:mendel}); men det er alts\aa{} ikke et s\aa{}dant bevis, som er form\aa{}let med dette afsnit.
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% figur 3 (30cm)
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,25cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/same.eps}}
\caption[Bevisstrukturen for lighedslemmaerne]{Bevisstrukturen for lighedslemmaerne. Kun lighedslemmaerne er vist. Figuren skal l\ae{}ses ligesom figur 2.}
\end{figure}
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\subsection{Par} \label{sec:parlig}
N\aa{}r vi skal vise et lighedslemma med konklusionen " [ bracket op2 metavar var x end metavar comma metavar var v end metavar end op2 zermelo is op2 metavar var y end metavar comma metavar var w end metavar end op2 end bracket ] ", er det ofte en god ide f\o{}rst at vise et ``delm\ae{}ngdelemma'' --- dvs.\ et lemma, der konkluderer, at " [ math op2 metavar var x end metavar comma metavar var v end metavar end op2 end math ] " er en delm\ae{}ngde af " [ math op2 metavar var y end metavar comma metavar var w end metavar end op2 end math ] ". Vi kan da vise lighedslemmaet ud fra " [ math lemma set equality suff condition end math ] " og to anvendelser af delm\ae{}ngdelemmaet. I dette underafsnit begynder vi med det f\o{}lgende delm\ae{}ngdelemma:
\begin{list}{}{
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\item " [ math define statement of lemma pair subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply metavar var s end metavar zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end define end math ] "
\item Beviset for " [ math lemma pair subset end math ] " kan gengives som f\o{}lger:
\begin{enumerate}
\item ``Antag " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ", " [ bracket metavar var v end metavar zermelo is metavar var w end metavar end bracket ] " og at " [ math metavar var s end metavar end math ] " tilh\o{}rer " [ math zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end math ] ". Der g\ae{}lder da " [ math metavar var s end metavar zermelo is metavar var x end metavar end math ] " eller " [ math metavar var s end metavar zermelo is metavar var y end metavar end math ] ".''
\item ``" [ math metavar var s end metavar zermelo is metavar var x end metavar end math ] " medf\o{}rer " [ bracket not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end bracket ] " (under antagelse af " [ bracket metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ").''
\item ``" [ math metavar var s end metavar zermelo is metavar var y end metavar end math ] " medf\o{}rer ogs\aa{} " [ bracket not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end bracket ] ".''
\item ``Derfor m\aa{} " [ bracket not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end bracket ] " g\ae{}lde ubetinget. Dette betyder igen, at " [ math metavar var s end metavar end math ] " tilh\o{}rer " [ math zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end math ] ", QED.''
\end{enumerate}
\item Punkt 2 og 3 varetages af de to lemmastumper " [ math lemma pair subset0 end math ] " og \\ \mbox{" [ math lemma pair subset1 end math ] "}. Punkt 1 og 4 vises i selve beviset for " [ math lemma pair subset end math ] ":
\item " [ math define statement of lemma pair subset0 as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end define end math ] "
\item " [ math define proof of lemma pair subset0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo is metavar var x end metavar infer lemma =transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var s end metavar zermelo is metavar var x end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var s end metavar zermelo is metavar var y end metavar cut prop lemma weaken or second modus ponens metavar var s end metavar zermelo is metavar var y end metavar conclude not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var w end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo is metavar var x end metavar infer not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar conclude quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma pair subset1 as system zf infer all metavar var s end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end define end math ] "
\item " [ math define proof of lemma pair subset1 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var v end metavar zermelo is metavar var w end metavar infer metavar var s end metavar zermelo is metavar var v end metavar infer lemma =transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus ponens metavar var s end metavar zermelo is metavar var v end metavar modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude metavar var s end metavar zermelo is metavar var w end metavar cut prop lemma weaken or first modus ponens metavar var s end metavar zermelo is metavar var w end metavar conclude not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut all metavar var s end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var v end metavar zermelo is metavar var w end metavar infer metavar var s end metavar zermelo is metavar var v end metavar infer not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar conclude quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma pair subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply metavar var s end metavar zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end define end math ] "
" [ math define proof of lemma pair subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair infer lemma pair2formula modus ponens metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair conclude not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var v end metavar cut lemma pair subset0 modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote conclude metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut 1rule mp modus ponens metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut lemma pair subset1 modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote conclude metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut 1rule mp modus ponens metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut prop lemma from disjuncts modus ponens not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var v end metavar modus ponens metavar var s end metavar zermelo is metavar var x end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar modus ponens metavar var s end metavar zermelo is metavar var v end metavar imply not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar conclude not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar cut lemma formula2pair modus ponens not0 metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo is metavar var w end metavar conclude metavar var s end metavar zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair infer metavar var s end metavar zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply metavar var s end metavar zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end quote state proof state cache var c end expand end define end math ] "
\item Kommet s\aa{} vidt kan vi bevise lighedslemmaet " [ math lemma same pair end math ] ". Vi viser, at " [ math zermelo pair metavar var x end metavar comma metavar var v end metavar end pair end math ] " er en delm\ae{}ngde af " [ math zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end math ] " (linie 8--11), og vice versa (linie 12--14):
\item " [ math define statement of lemma same pair as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var v end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer zermelo pair metavar var x end metavar comma metavar var v end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end define end math ] "
\item " [ math define proof of lemma same pair as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var v end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer lemma pair subset modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote conclude metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair cut prop lemma mp2 modus ponens metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var y end metavar zermelo is metavar var x end metavar cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude metavar var w end metavar zermelo is metavar var v end metavar cut lemma pair subset modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote conclude metavar var y end metavar zermelo is metavar var x end metavar imply metavar var w end metavar zermelo is metavar var v end metavar imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair cut prop lemma mp2 modus ponens metavar var y end metavar zermelo is metavar var x end metavar imply metavar var w end metavar zermelo is metavar var v end metavar imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair modus ponens metavar var y end metavar zermelo is metavar var x end metavar modus ponens metavar var w end metavar zermelo is metavar var v end metavar conclude object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair cut lemma set equality suff condition(t) modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var v end metavar end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end quote modus ponens object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair modus ponens object var var t end var zermelo in zermelo pair metavar var y end metavar comma metavar var w end metavar end pair imply object var var t end var zermelo in zermelo pair metavar var x end metavar comma metavar var v end metavar end pair conclude zermelo pair metavar var x end metavar comma metavar var v end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var w end metavar end pair end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Singleton-m\ae{}ngde}
Lighedslemmaet for singleton-konstruktionen ser s\aa{}ledes ud:
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\item " [ math define statement of lemma same singleton as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end define end math ] "
\item Da " [ bracket zermelo pair var x comma var x end pair end bracket ] " er makrodefineret som " [ bracket zermelo pair var x comma var x end pair end bracket ] ", kan vi let vise " [ math lemma same singleton end math ] " ud fra " [ math lemma same pair end math ] ":
\item " [ math define proof of lemma same singleton as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer lemma same pair modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair cut 1rule repetition modus ponens zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair conclude zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Foreningsm\ae{}ngde}
Delm\ae{}ngdelemmaet mht.\ " [ bracket union var x end union end bracket ] " ser s\aa{}ledes ud:
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\item " [ math define statement of lemma union subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union imply metavar var s end metavar zermelo in union metavar var y end metavar end union end define end math ] "
\item Bevisets kerne (linie 7--11) er en simpel anvendelse af definitionen af " [ bracket union var x end union end bracket ] ":
\item " [ math define proof of lemma union subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo in union metavar var x end metavar end union infer lemma union2formula modus ponens metavar var s end metavar zermelo in union metavar var x end metavar end union conclude not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in existential var var j end var cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar conclude existential var var j end var zermelo in metavar var x end metavar cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens existential var var j end var zermelo in metavar var x end metavar conclude existential var var j end var zermelo in metavar var y end metavar cut lemma formula2union modus ponens metavar var s end metavar zermelo in existential var var j end var modus ponens existential var var j end var zermelo in metavar var y end metavar conclude metavar var s end metavar zermelo in union metavar var y end metavar end union cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var s end metavar zermelo in union metavar var x end metavar end union infer metavar var s end metavar zermelo in union metavar var y end metavar end union conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union imply metavar var s end metavar zermelo in union metavar var y end metavar end union end quote state proof state cache var c end expand end define end math ] "
\item I beviset for lighedslemmaet " [ math lemma same union end math ] " viser vi, at " [ math union metavar var x end metavar end union end math ] " er en delm\ae{}ngde af " [ math union metavar var y end metavar end union end math ] " (linie 4--6), og vice versa (line 7--9):
\item " [ math define statement of lemma same union as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer union metavar var x end metavar end union zermelo is union metavar var y end metavar end union end define end math ] "
\item " [ math define proof of lemma same union as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer lemma union subset modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote conclude metavar var x end metavar zermelo is metavar var y end metavar imply object var var s end var zermelo in union metavar var x end metavar end union imply object var var s end var zermelo in union metavar var y end metavar end union cut 1rule mp modus ponens metavar var x end metavar zermelo is metavar var y end metavar imply object var var s end var zermelo in union metavar var x end metavar end union imply object var var s end var zermelo in union metavar var y end metavar end union modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude object var var s end var zermelo in union metavar var x end metavar end union imply object var var s end var zermelo in union metavar var y end metavar end union cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var y end metavar zermelo is metavar var x end metavar cut lemma union subset modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote conclude metavar var y end metavar zermelo is metavar var x end metavar imply object var var s end var zermelo in union metavar var y end metavar end union imply object var var s end var zermelo in union metavar var x end metavar end union cut 1rule mp modus ponens metavar var y end metavar zermelo is metavar var x end metavar imply object var var s end var zermelo in union metavar var y end metavar end union imply object var var s end var zermelo in union metavar var x end metavar end union modus ponens metavar var y end metavar zermelo is metavar var x end metavar conclude object var var s end var zermelo in union metavar var y end metavar end union imply object var var s end var zermelo in union metavar var x end metavar end union cut lemma set equality suff condition modus ponens object var var s end var zermelo in union metavar var x end metavar end union imply object var var s end var zermelo in union metavar var y end metavar end union modus ponens object var var s end var zermelo in union metavar var y end metavar end union imply object var var s end var zermelo in union metavar var x end metavar end union conclude union metavar var x end metavar end union zermelo is union metavar var y end metavar end union end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Bin\ae{}r foreningsm\ae{}ngde} \label{sec:binaerforen}
I lighedslemmaet for " [ bracket union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union end bracket ] " bliver problemet med sidebetingelserne meget tydeligt:
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\item" [ math define statement of lemma same binary union as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end union zermelo is union zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end union end define end math ] "
\end{list}
Alt i alt er der 16 sidebetingelser. Et s\aa{}dant lemma er jo ikke til at arbejde med. Derfor vil vi bruge objektvariable i formuleringen af de lemmaer, der afh\ae{}nger af " [ math lemma same binary union end math ] ". P\aa{} d\'{e}n m\aa{}de bliver alle sidebetingelserne automatisk opfyldt, og vi beh\o{}ver ikke at behandle dem eksplicit.
Problemet med " [ math lemma same binary union end math ] " er, at det nedarver sidebetingelser fra tre forskellige lemmaer (jvf.\ figur 3), og at der ikke er noget overlap mellem sidebetingelserne. Mange af sidebetingelserne er ensbetydende; f.eks.\ medf\o{}rer " [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote end bracket ] " og " [ bracket quote object var var s end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote end bracket ] " hinanden. Bevischeckeren er imidlertid ikke i stand til at udnytte dette; vi kan ikke g\o{}re andet end at opremse alle sidebetingelserne.
Ironisk nok er selve beviset for " [ math lemma same binary union end math ] " ret enkelt. Vi s\ae{}tter blot lighedslemmaerne " [ math lemma same singleton end math ] ", " [ math lemma same pair end math ] " og " [ math lemma same union end math ] " sammen:
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\item " [ math define proof of lemma same binary union as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote endorse quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote endorse quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer lemma same singleton modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair cut lemma same singleton modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote modus ponens metavar var v end metavar zermelo is metavar var w end metavar conclude zermelo pair metavar var v end metavar comma metavar var v end metavar end pair zermelo is zermelo pair metavar var w end metavar comma metavar var w end metavar end pair cut lemma same pair modus probans quote object var var s end var end quote avoid zero quote zermelo pair metavar var x end metavar comma metavar var x end metavar end pair end quote modus probans quote object var var s end var end quote avoid zero quote zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end quote modus probans quote object var var s end var end quote avoid zero quote zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end quote modus probans quote object var var s end var end quote avoid zero quote zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote modus probans quote object var var t end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote modus ponens zermelo pair metavar var x end metavar comma metavar var x end metavar end pair zermelo is zermelo pair metavar var y end metavar comma metavar var y end metavar end pair modus ponens zermelo pair metavar var v end metavar comma metavar var v end metavar end pair zermelo is zermelo pair metavar var w end metavar comma metavar var w end metavar end pair conclude zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair zermelo is zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair cut lemma same union modus probans quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end quote modus probans quote object var var s end var end quote avoid zero quote zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end quote modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair zermelo is zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair conclude union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end union zermelo is union zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end union cut 1rule repetition modus ponens union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end union zermelo is union zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end union conclude union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var v end metavar comma metavar var v end metavar end pair end pair end union zermelo is union zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var w end metavar comma metavar var w end metavar end pair end pair end union end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Separation}
Delm\ae{}ngdelemmaet for konstruktionen " [ bracket the set of ph in metavar var t end metavar such that metavar var f end metavar end set end bracket ] " ser s\aa{}ledes ud:
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\item " [ math define statement of lemma separation subset as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set end define end math ] "
\item I beviset g\aa{}r vi fra " [ bracket metavar var s end metavar zermelo in metavar var x end metavar end bracket ] " til " [ bracket metavar var s end metavar zermelo in metavar var y end metavar end bracket ] " (linie 9--10), og fra " [ bracket metavar var a end metavar end bracket ] " til " [ bracket metavar var b end metavar end bracket ] " (linie 11--12):
\item " [ math define proof of lemma separation subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set infer lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var a end metavar cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var a end metavar conclude metavar var s end metavar zermelo in metavar var x end metavar cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var a end metavar conclude metavar var a end metavar cut prop lemma iff second modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut lemma formula2separation modus ponens metavar var s end metavar zermelo in metavar var y end metavar modus ponens metavar var b end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set infer metavar var s end metavar zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set end quote state proof state cache var c end expand end define end math ] "
\item I beviset for lighedslemmaet " [ math lemma same separation end math ] " viser vi, at " [ math the set of ph in metavar var x end metavar such that metavar var a end metavar end set end math ] " er en delm\ae{}ngde af " [ math the set of ph in metavar var y end metavar such that metavar var b end metavar end set end math ] " (linie 4--7), og vice versa (linie 8--11):
\item " [ math define statement of lemma same separation as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer the set of ph in metavar var x end metavar such that metavar var a end metavar end set zermelo is the set of ph in metavar var y end metavar such that metavar var b end metavar end set end define end math ] "
\item " [ math define proof of lemma same separation as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer lemma separation subset modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote conclude metavar var x end metavar zermelo is metavar var y end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set cut prop lemma mp2 modus ponens metavar var x end metavar zermelo is metavar var y end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set cut lemma =symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar conclude metavar var y end metavar zermelo is metavar var x end metavar cut prop lemma iff commutativity modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut lemma separation subset modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote conclude metavar var y end metavar zermelo is metavar var x end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set cut prop lemma mp2 modus ponens metavar var y end metavar zermelo is metavar var x end metavar imply not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set modus ponens metavar var y end metavar zermelo is metavar var x end metavar modus ponens not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar conclude object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set cut lemma set equality suff condition modus ponens object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set modus ponens object var var s end var zermelo in the set of ph in metavar var y end metavar such that metavar var b end metavar end set imply object var var s end var zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set conclude the set of ph in metavar var x end metavar such that metavar var a end metavar end set zermelo is the set of ph in metavar var y end metavar such that metavar var b end metavar end set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Bin\ae{}r f\ae{}llesm\ae{}ngde} \label{sec:sameinter}
Delm\ae{}ngdelemmaet for konstruktionen " [ bracket the set of ph in union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union such that not0 placeholder-var var c end var zermelo in var x imply not0 placeholder-var var c end var zermelo in var y end set end bracket ] " adskiller sig fra normen ved ikke at n\ae{}vne " [ bracket the set of ph in union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union such that not0 placeholder-var var c end var zermelo in var x imply not0 placeholder-var var c end var zermelo in var y end set end bracket ] " eksplicit:
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\item " [ math define statement of lemma intersection subset as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var w end metavar end define end math ] "
\item Bevisets kerne best\aa{}r af to anvendelser af " [ math lemma set equality nec condition end math ] " (linie 10--11 og \mbox{12--13}):
\item " [ math define proof of lemma intersection subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar infer prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar conclude metavar var s end metavar zermelo in metavar var x end metavar cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote modus ponens metavar var x end metavar zermelo is metavar var y end metavar modus ponens metavar var s end metavar zermelo in metavar var x end metavar conclude metavar var s end metavar zermelo in metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar conclude metavar var s end metavar zermelo in metavar var v end metavar cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote modus ponens metavar var v end metavar zermelo is metavar var w end metavar modus ponens metavar var s end metavar zermelo in metavar var v end metavar conclude metavar var s end metavar zermelo in metavar var w end metavar cut prop lemma join conjuncts modus ponens metavar var s end metavar zermelo in metavar var y end metavar modus ponens metavar var s end metavar zermelo in metavar var w end metavar conclude not0 metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var w end metavar cut all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed 1rule deduction modus ponens all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar infer metavar var v end metavar zermelo is metavar var w end metavar infer not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar infer not0 metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var w end metavar conclude quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var y end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var v end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var w end metavar end quote endorse metavar var x end metavar zermelo is metavar var y end metavar imply metavar var v end metavar zermelo is metavar var w end metavar imply not0 metavar var s end metavar zermelo in metavar var x end metavar imply not0 metavar var s end metavar zermelo in metavar var v end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
S\aa{} er vi kommet til lighedslemmaet " [ math lemma same intersection end math ] ". Da dette lemma afh\ae{}nger af " [ math lemma same binary union end math ] ", sl\aa{}r vi her over til objektvariable (jvf.\ afsnit \ref{sec:binaerforen}). Den store ulempe herved er, at objektvariable ikke kan instantieres til andre variable. Derfor vil en formulering som f.eks.
" [ bracket object var var x end var zermelo is object var var y end var infer object var var v end var zermelo is object var var w end var infer the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var v end var comma object var var v end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var v end var end set zermelo is the set of ph in union zermelo pair zermelo pair object var var y end var comma object var var y end var end pair comma zermelo pair object var var w end var comma object var var w end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in object var var w end var end set end bracket ] "
sandsynligvis ikke kunne anvendes senere hen i rapporten, da vi ikke kan instantiere " [ math object var var x end var end math ] ", " [ math object var var y end var end math ] ", " [ math object var var v end var end math ] " eller " [ math object var var w end var end math ] ".
Vi m\aa{} alts\aa{} unders\o{}ge, hvad " [ math lemma same intersection end math ] " egentlig skal bruges til, f\o{}r vi formulerer lemmaet. Det viser sig, at den f\o{}lgende formulering kan bruges:
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\item " [ math define statement of lemma same intersection as system zf infer object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set infer object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set infer the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set end define end math ] "
\item Man kan sige, at vi har instantieret objektvariablene ``p\aa{} forh\aa{}nd''.
\item Beviset for " [ math lemma same intersection end math ] " kan beskrives som f\o{}lger:
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\item
\begin{enumerate}
\item Fra " [ math lemma same binary union end math ] " f\aa{}r vi, at de to bin\ae{}re foreningsm\ae{}ngder er \'{e}ns (linie 3).
\item De to formler " [ bracket not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end bracket ] " og \\ \mbox{" [ bracket not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end bracket ] "} implicerer hinanden (linie 4--5 og 6--9).
\item Fra punkt 1, punkt 2 og " [ math lemma same separation end math ] " f\aa{}r vi da, at de to bin\ae{}re f\ae{}llesm\ae{}ngder er \'{e}ns (linie 10--12).
\end{enumerate}
%
\item Her er beviset:
\item " [ math define proof of lemma same intersection as lambda var c dot lambda var x dot proof expand quote system zf infer object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set infer object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set infer lemma same binary union modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set modus ponens object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union zermelo is union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union cut lemma intersection subset conclude object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set cut prop lemma mp2 modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set modus ponens object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set cut lemma =symmetry modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set conclude the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var cut lemma =symmetry modus ponens object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var cut lemma intersection subset conclude the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var imply the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var cut prop lemma mp2 modus ponens the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var imply the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var modus ponens the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var modus ponens the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var conclude not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var cut prop lemma join conjuncts modus ponens not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set modus ponens not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var conclude not0 not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var cut lemma same separation modus ponens union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union zermelo is union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union modus ponens not0 not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var conclude the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set cut 1rule repetition modus ponens the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set conclude the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\section{Hovedresultatet} \label{sec:hoved}
Vi kan nu endelig g\aa{} i gang med hovedresultatet, som vi allerede udtrykte formelt i afsnit \ref{sec:partition}:
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\item " [ math define statement of theorem eq-system is partition as system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var end define end math ] "
\end{list}
Vi skal vise tre ting:
\begin{itemize}
\item Ingen \ae{}kvivalensklasser er tomme (underafsnit \ref{sec:nummer1}),
\item Alle \ae{}kvivalensklasser er disjunkte (underafsnit \ref{sec:nummer2}), og
\item F\ae{}llesm\ae{}ngden af alle \ae{}kvivalensklasserne er lig den oprindelige m\ae{}ngde " [ math object var var big set end var end math ] " (underafsnit \ref{sec:nummer3}).
\end{itemize}
\noindent Til sidst binder vi alle tr\aa{}dene sammen i underafsnit \ref{sec:sidste}.
\subsection{Ingen \ae{}kvivalensklasser er tomme} \label{sec:nummer1}
Lemmaet " [ math lemma auto member end math ] " udsiger, at \ae{}kvivalensklassen " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " indeholder " [ math metavar var s end metavar end math ] " selv som medlem:
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\item " [ math define statement of lemma auto member as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item " [ math lemma auto member end math ] " f\o{}lger af, at vi har " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end math ] ", fordi \ae{}kvivalensrelationen " [ math metavar var r end metavar end math ] " er refleksiv:
\item " [ math define proof of lemma auto member as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer lemma er is reflexive modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut lemma reflexivity modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar cut lemma formula2separation modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define end math ] "
\item Ud fra " [ math lemma auto member end math ] " kan vi nu vise, at ingen medlemmer af et \ae{}kvivalenssystem er tomme:
\item " [ math define statement of lemma eq-system not empty as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set end define end math ] "
\item Pga.\ sidebetingelserne deler vi beviset op i to, med " [ math lemma eq-system not empty0 end math ] " som lemmastump:
\item " [ math define statement of lemma eq-system not empty0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set end define end math ] "
\item N\ae{}sten hele beviset varetages af " [ math lemma eq-system not empty0 end math ] ". Beviset forl\o{}ber som f\o{}lger: Lad " [ math metavar var s end metavar end math ] " v\ae{}re et medlem af et \ae{}kvivalenssystem (linie 8). Pr. definition er " [ math metavar var s end metavar end math ] " en \ae{}kvivalensklasse (linie 9--12). Da enhver \ae{}kvivalensklasse har et medlem, har " [ math metavar var s end metavar end math ] " et medlem (linie 13--14). Da enhver m\ae{}ngde med et medlem ikke er tom, er " [ math metavar var s end metavar end math ] " ikke tom, QED (linie 15).
\item " [ math define proof of lemma eq-system not empty0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude not0 metavar var s end metavar zermelo in power metavar var big set end metavar end power imply not0 not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in power metavar var big set end metavar end power imply not0 not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar conclude not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar cut prop lemma first conjunct modus ponens not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar conclude existential var var a end var zermelo in metavar var big set end metavar cut prop lemma second conjunct modus ponens not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar cut lemma auto member modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens existential var var a end var zermelo in metavar var big set end metavar conclude existential var var a end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set cut lemma set equality nec condition modus probans quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is metavar var s end metavar modus ponens existential var var a end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set conclude existential var var a end var zermelo in metavar var s end metavar cut lemma member not empty modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus ponens existential var var a end var zermelo in metavar var s end metavar conclude not0 metavar var s end metavar zermelo is zermelo empty set cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var s end metavar zermelo is zermelo empty set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item I hovedlemmaet skifter vi en implikation ud med en inferens:
\item " [ math define statement of lemma eq-system not empty as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma eq-system not empty as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer lemma eq-system not empty0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var s end var end quote avoid zero quote the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set cut 1rule mp modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var s end metavar zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Alle \ae{}kvivalensklasser er disjunkte} \label{sec:nummer2}
Dette underafsnit indeholder den mest komplicerede del af beviset for hovedresultatet. Som det kan ses af figur 4, har jeg m\aa{}ttet tage et ``snyde-aksiom'' til hj\ae{}lp for at blive f\ae{}rdig.
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% figur 4
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics[0cm,5cm][20cm,30cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/nummer2.eps}}
\caption[Bevisstrukturen for underafsnit \ref{sec:nummer2}]{Bevisstrukturen for underafsnit \ref{sec:nummer2}. Kun lemmaerne fra dette afsnit er vist. Figuren skal l\ae{}ses ligesom figur 2. Kassen m\ae{}rket ``" [ math cheating axiom all disjoint end math ] "'' repr\ae{}senterer et ``snyde-aksiom'', s\aa{} der g\aa{}r ingen pile ind i denne kasse.}
\end{figure}
\clearpage
\subsubsection{Lemmaet ``" [ math lemma eq subset end math ] "''}
Indholdet af det f\o{}rste lemma, " [ math lemma eq subset end math ] ", kan udtrykkes som f\o{}lger: ``Lad " [ math metavar var x end metavar end math ] " og " [ math metavar var y end metavar end math ] " v\ae{}re to medlemmer af " [ math metavar var big set end metavar end math ] ". Hvis der g\ae{}lder " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ", s\aa{} er \ae{}kvivalensklassen " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " en delm\ae{}ngde af \ae{}kvivalensklassen " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] "'':
\begin{list}{}{
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\item " [ math define statement of lemma eq subset as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item Beviset forl\o{}ber som f\o{}lger:
%
\begin{enumerate}
\item Lad " [ math metavar var s end metavar end math ] " v\ae{}re et vilk\aa{}rligt medlem af " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] ". Vi har da " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end math ] " \\
\mbox{(linie 13--16)}.
\item Ud fra " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end math ] "
og antagelsen " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] "
f\aa{}r vi " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ", da " [ math metavar var r end metavar end math ] " er transitiv (linie 17--19).
\item " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] " medf\o{}rer igen " [ math metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] ", QED (linie 20).
\end{enumerate}
%
\item Hele dette r\ae{}sonnement gennemf\o{}res i lemmastumpen " [ math lemma eq subset0 end math ] ". Selve beviset for " [ math lemma eq subset end math ] " best\aa{}r af den s\ae{}dvanlige omskrivning fra implikation til inferens. Her er beviserne:
\item " [ math define statement of lemma eq subset0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item " [ math define proof of lemma eq subset0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set infer 1rule repetition modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut lemma er is transitive modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in metavar var big set end metavar cut lemma transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut lemma formula2separation modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of lemma eq subset as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item " [ math define proof of lemma eq subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer lemma eq subset0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma mp4 modus ponens metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Lemmaet ``" [ math lemma equivalence nec condition end math ] "''}
Det er nu en let sag at vise den st\ae{}rkere p\aa{}stand, at " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " rent faktisk er lig med " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] "\footnote{Dette er lemma 4.4.a.1 i \cite{kn:hrba}.}. Som s\aa{} ofte f\o{}r ligger den interessante del af beviset i en lemmastump, der her hedder ``" [ math lemma equivalence nec condition0 end math ] "'':
\begin{list}{}{
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\item " [ math define statement of lemma equivalence nec condition0 as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item Beviset kan beskrives som f\o{}lger:
\begin{enumerate}
\item Fra " [ math lemma eq subset end math ] " ved vi, at antagelsen " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] " medf\o{}rer, at " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " er en delm\ae{}ngde af " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] ". (Linie 13).
\item Da " [ math metavar var r end metavar end math ] " er symmetrisk, g\ae{}lder der ogs\aa{} " [ math zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end math ] ". (Linie 14--15).
\item Vi kan derfor bruge " [ math lemma eq subset end math ] " \'{e}n gang til --- vi f\aa{}r, at " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " er en delm\ae{}ngde af " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] ". (Linie 16).
\item Ud fra punkt 1 og 3 f\o{}lger " [ bracket the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end bracket ] ", QED. (Linie 17).
\end{enumerate}
\item Her er selve beviset:
\item " [ math define proof of lemma equivalence nec condition0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer lemma eq subset modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma er is symmetric modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut lemma symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut lemma eq subset modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma set equality suff condition modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define end math ] "
\item Vi mangler nu blot at skifte fra implikationer til inferenser:
\item " [ math define statement of lemma equivalence nec condition as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define end math ] "
\item " [ math define proof of lemma equivalence nec condition as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer lemma equivalence nec condition0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma mp3 modus ponens metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Lemmaet ``" [ math lemma none-equivalence nec condition end math ] "''}
Resultatet fra dette under-underafsnit kan formuleres som f\o{}lger: ``Lad " [ math metavar var x end metavar end math ] " og " [ math metavar var y end metavar end math ] " v\ae{}re to medlemmer af
" [ math metavar var big set end metavar end math ] ", som ikke opfylder " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ". Da er de to \ae{}kvivalensklasser " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " og " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " indbyrdes disjunkte.''\footnote{Dette er lemma 4.4.b.1 i \cite{kn:hrba}.}
Vi viser dette ved at vise den kontrapositive p\aa{}stand: Hvis \ae{}kvivalensklasserne har et medlem tilf\ae{}lles, s\aa{} g\ae{}lder " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ". Dette varetages af lemmaet " [ math lemma none-equivalence nec condition0 end math ] ":
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\item " [ math define statement of lemma none-equivalence nec condition0 as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end define end math ] "
\item Beviset for " [ math lemma equivalence nec condition0 end math ] " kan beskrives som f\o{}lger:
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\begin{enumerate}
\item Antag at " [ math metavar var s end metavar end math ] " tilh\o{}rer b\aa{}de " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " og " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] ". Da g\ae{}lder " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end math ] " (linie 17--19) og " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] " (linie 20--22).
\item Da " [ math metavar var r end metavar end math ] " er symmetrisk, f\o{}lger " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end math ] " af " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end math ] ". (Linie 24).
\item Da " [ math metavar var r end metavar end math ] " ogs\aa{} er transitiv, medf\o{}rer " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end math ] " og " [ math zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] " tilsammen " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ", QED. (Linie 25).
\end{enumerate}
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Her er selve beviset:
\item " [ math define proof of lemma none-equivalence nec condition0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set infer lemma er is symmetric modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut lemma er is transitive modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar cut lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set conclude not0 metavar var s end metavar zermelo in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union imply not0 not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union imply not0 not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in metavar var big set end metavar cut lemma symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar cut lemma transitivity modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end quote state proof state cache var c end expand end define end math ] "
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\\ \mbox{" [ math lemma none-equivalence nec condition0 end math ] "}. Dette sker i beviset for en lemmastump ved navn " [ math lemma none-equivalence nec condition1 end math ] ":
\item " [ math define statement of lemma none-equivalence nec condition1 as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma none-equivalence nec condition1 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer lemma none-equivalence nec condition0 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma mp3 modus ponens metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma mt modus ponens object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar modus ponens not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude not0 object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set cut lemma unique empty set modus ponens not0 object var var s end var zermelo in the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set conclude the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set cut all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item I beviset for hovedlemmaet " [ math lemma none-equivalence nec condition end math ] " foretager vi den s\ae{}dvanlige konvertering fra " [ bracket var x imply var y end bracket ] " til " [ bracket var x infer var y end bracket ] ":
\item " [ math define statement of lemma none-equivalence nec condition as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma none-equivalence nec condition as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer lemma none-equivalence nec condition1 modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set cut prop lemma mp3 modus ponens metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{To \ae{}kvivalensklasser er disjunkte}
Vi kan nu bevise, at to forskellige \ae{}kvivalensklasser er disjunkte:
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\item " [ math define statement of lemma equivalence classes are disjoint as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set infer the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end define end math ] "
\item Beviset for " [ math lemma equivalence classes are disjoint end math ] " forl\o{}ber som f\o{}lger:
%
\begin{enumerate}
\item Vi antager, at " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " og " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " ikke er lig med hinanden. Fra " [ math lemma equivalence nec condition end math ] " ved vi, at hvis der gjaldt " [ math zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ", s\aa{} ville " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " og " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " v\ae{}re lig med hinanden. Derfor m\aa{} der g\ae{}lde " [ math not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] ". \mbox{(Linie 12--13)}.
\item Fra " [ math lemma none-equivalence nec condition end math ] " ved vi, at " [ math not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end math ] " medf\o{}rer, at de to \ae{}kvivalensklasser er disjunkte; QED. (Linie 14--15).
\end{enumerate}
%
Her er beviset:
\item " [ math define proof of lemma equivalence classes are disjoint as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set infer lemma equivalence nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma mt modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar cut lemma none-equivalence nec condition modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set cut 1rule mp modus ponens not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set modus ponens not0 zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude the set of ph in union zermelo pair zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end pair comma zermelo pair the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set comma the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply not0 placeholder-var var c end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end set zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Alle \ae{}kvivalensklasser er disjunkte}
\label{sec:alldisjoint}
Ud fra " [ math lemma equivalence classes are disjoint end math ] " kan vi nu vise, at alle medlemmer af et \ae{}kvivalenssystem er parvis disjunkte. Da beviset afh\ae{}nger af " [ math lemma same intersection end math ] ", som igen afh\ae{}nger af " [ math lemma same binary union end math ] ", bruger vi objektvariable (jvf.\ afsnit \ref{sec:binaerforen}):
\begin{list}{}{
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\item " [ math define statement of lemma all disjoint as system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set infer object var var y end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set infer not0 object var var x end var zermelo is object var var y end var infer the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is zermelo empty set end define end math ] "
\item Beviset forl\o{}ber som f\o{}lger:
%
\begin{enumerate}
\item Lad " [ math object var var x end var end math ] " og " [ math object var var y end var end math ] " v\ae{}re to forskellige medlemmer af et \ae{}kvivalenssystem " [ math the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end math ] " (linie 2--4).
\item M\ae{}ngden " [ math object var var x end var end math ] " m\aa{} v\ae{}re en \ae{}kvivalensklasse; vi kan alts\aa{} skrive \\
" [ math object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end math ] " for et " [ math existential var var a end var end math ] ", som tilh\o{}rer " [ math object var var big set end var end math ] ". (Linie 5--9).
\item Det samme g\ae{}lder for " [ math object var var y end var end math ] "; her skriver vi
" [ math object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end math ] ". (Linie 10--14).
\item Da " [ math object var var x end var end math ] " og " [ math object var var y end var end math ] " er forskellige, m\aa{} " [ math the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end math ] " og " [ math the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end math ] " ogs\aa{} v\ae{}re forskellige. (Linie 15).
\item Heraf f\o{}lger, at f\ae{}llesm\ae{}ngden mellem " [ math the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end math ] " og " [ math the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end math ] " er tom. (Linie 16).
\item Af lighedslemmaet " [ math lemma same intersection end math ] " f\aa{}r vi \\ \mbox{" [ math the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end set end math ] "}. (Linie 17).
\item Da " [ math the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end set end math ] " er tom (punkt 5), m\aa{} " [ math the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set end math ] " derfor ogs\aa{} v\ae{}re tom, QED. (Linie 18).
\end{enumerate}
%
Her er beviset:
\item " [ math define proof of lemma all disjoint as lambda var c dot lambda var x dot proof expand quote system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set infer object var var y end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set infer not0 object var var x end var zermelo is object var var y end var infer lemma separation2formula modus ponens object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set conclude not0 object var var x end var zermelo in power object var var big set end var end power imply not0 not0 existential var var a end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var cut prop lemma second conjunct modus ponens not0 object var var x end var zermelo in power object var var big set end var end power imply not0 not0 existential var var a end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var conclude not0 existential var var a end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var cut prop lemma first conjunct modus ponens not0 existential var var a end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var conclude existential var var a end var zermelo in object var var big set end var cut prop lemma second conjunct modus ponens not0 existential var var a end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var conclude the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var cut lemma =symmetry modus ponens the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is object var var x end var conclude object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set cut lemma separation2formula modus ponens object var var y end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set conclude not0 object var var y end var zermelo in power object var var big set end var end power imply not0 not0 existential var var b end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var cut prop lemma second conjunct modus ponens not0 object var var y end var zermelo in power object var var big set end var end power imply not0 not0 existential var var b end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var conclude not0 existential var var b end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var cut prop lemma first conjunct modus ponens not0 existential var var b end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var conclude existential var var b end var zermelo in object var var big set end var cut prop lemma second conjunct modus ponens not0 existential var var b end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var conclude the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var cut lemma =symmetry modus ponens the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set zermelo is object var var y end var conclude object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set cut lemma transfer ~is modus ponens not0 object var var x end var zermelo is object var var y end var modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set modus ponens object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set cut lemma equivalence classes are disjoint modus ponens existential var var a end var zermelo in object var var big set end var modus ponens existential var var b end var zermelo in object var var big set end var modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var modus ponens not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set zermelo is zermelo empty set cut lemma same intersection modus ponens object var var x end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set modus ponens object var var y end var zermelo is the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set conclude the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set cut lemma =transitivity modus ponens the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set modus ponens the set of ph in union zermelo pair zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set end pair comma zermelo pair the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set comma the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end pair end pair end union such that not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in object var var r end var end set imply not0 placeholder-var var c end var zermelo in the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var b end var end pair end pair zermelo in object var var r end var end set end set zermelo is zermelo empty set conclude the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item Bem\ae{}rk at vi i dette bevis introducerer b\aa{}de " [ math existential var var a end var end math ] " og " [ math existential var var b end var end math ] " ud fra definitionen af " [ bracket the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end bracket ] ", selvom denne definition bruger eksistensvariablen " [ math existential var var t end var end math ] ". Vi nyder her gavn af, at separationsaksiomet tillader introduktion af ubrugte eksistens-variable (som n\ae{}vnt i afsnit \ref{sec:separation}).
\end{list}
\subsubsection{Implikation i stedet for inferens} \label{sec:konvertering}
Det sidste skridt i dette afsnit er at skifte inferenserne fra " [ math lemma all disjoint end math ] " ud med implikationer. (Dette er p\aa{}kr\ae{}vet for, at resultatet fra " [ math lemma all disjoint end math ] " kan bruges i beviset for hovedresultatet). Vi plejer jo at bruge deduktionsreglen til d\'{e}n slags konverteringer; men der et problem. Vi kan ikke konkludere
\display{
" [ bracket not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var imply object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var y end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var x end var zermelo is object var var y end var imply the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is zermelo empty set end bracket ] "}
\noindent ud fra " [ math 1rule deduction end math ] "; vi kan kun konkludere
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\display{" [ bracket for all objects object var var r end var indeed for all objects object var var big set end var indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var imply for all objects object var var r end var indeed for all objects object var var x end var indeed for all objects object var var big set end var indeed object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply for all objects object var var r end var indeed for all objects object var var y end var indeed for all objects object var var big set end var indeed object var var y end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply for all objects object var var x end var indeed for all objects object var var y end var indeed not0 object var var x end var zermelo is object var var y end var imply the set of ph in union zermelo pair zermelo pair object var var x end var comma object var var x end var end pair comma zermelo pair object var var y end var comma object var var y end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var x end var imply not0 placeholder-var var c end var zermelo in object var var y end var end set zermelo is zermelo empty set end bracket ] ".}
\noindent Deduktionsreglen tillader alts\aa{} ikke frie objektvariable i antecedenterne. Dette er ogs\aa{} et rimeligt krav; ellers ville vi f.eks.\ kunne slutte den falske implikation " [ bracket object var var s end var zermelo is zermelo empty set imply for all objects object var var s end var indeed object var var s end var zermelo is zermelo empty set end bracket ] " ud fra lemmaet " [ bracket object var var s end var zermelo is zermelo empty set infer for all objects object var var s end var indeed object var var s end var zermelo is zermelo empty set end bracket ] ". Problemet er, at beviset for hovedresultatet kr\ae{}ver, at implikationen er renset for objektkvantorer. S\aa{} vi er alts\aa{} havnet i en blindgyde.
Vi l\o{}ser problemet ved at indf\o{}re f\o{}lgende ``snyde-aksiom'':
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\display{" [ math define statement of cheating axiom all disjoint as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var x end metavar zermelo is metavar var y end metavar infer the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set end define , define proof of cheating axiom all disjoint as rule tactic end define end math ] "}
\noindent I " [ math cheating axiom all disjoint end math ] " formulerer vi " [ math lemma all disjoint end math ] " ved hj\ae{}lp af metavariable. Vi kan opfatte " [ math cheating axiom all disjoint end math ] " som d\'{e}t resultat, vi var n\aa{}et frem til, hvis vi havde k\ae{}mpet videre med de mange sidebetingelser (bortset fra at der ikke er nogen sidebetingelser i formuleringen af " [ math cheating axiom all disjoint end math ] "). Ud fra denne formulering af " [ math lemma all disjoint end math ] " er det nemt at aflede en implikation:
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\item " [ math define statement of lemma all disjoint-imply as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var x end metavar zermelo is metavar var y end metavar imply the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set end define end math ] "
\item " [ math define proof of lemma all disjoint-imply as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var x end metavar zermelo is metavar var y end metavar infer cheating axiom all disjoint modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set modus ponens metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set modus ponens not0 metavar var x end metavar zermelo is metavar var y end metavar conclude the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set cut all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var x end metavar zermelo is metavar var y end metavar infer the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set imply not0 metavar var x end metavar zermelo is metavar var y end metavar imply the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set end quote state proof state cache var c end expand end define end math ] "
\item Vores brug af metavariable betyder, at sidebetingelserne straks vender tilbage --- dog i et t\aa{}leligt omfang. Det er lemmaet " [ math lemma all disjoint-imply end math ] ", som vi kommer til at bruge i beviset for " [ math theorem eq-system is partition end math ] ".
\end{list}
\subsection{\AE{}kvivalensklassernes foreningsm\ae{}ngde}
\label{sec:nummer3}
I dette underafsnit viser vi f\o{}rst, at " [ math metavar var big set end metavar end math ] " er en delm\ae{}ngde af \ae{}kvivalensklassernes foreningsm\ae{}ngde " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] " (under-underafsnit \ref{sec:union1}), og vice versa (under-underafsnit \ref{sec:union2}). Herudfra kan vi let vise, at de to m\ae{}ngder er lig hinanden (under-underafsnit \ref{sec:union3}).
\subsubsection{Den ene halvdel} \label{sec:union1}
Lemmaet " [ math lemma equivalence class is subset end math ] " udsiger, at en \ae{}kvivalensklasse " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " defineret p\aa{} en m\ae{}ngde " [ math metavar var big set end metavar end math ] " er en delm\ae{}ngde af " [ math metavar var big set end metavar end math ] ":
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\item " [ math define statement of lemma equivalence class is subset as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var big set end metavar indeed metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in metavar var big set end metavar end define end math ] "
\item Lemmaet f\o{}lger direkte af definitionen af " [ bracket the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set end bracket ] " fra afsnit \ref{sec:eqclass}:
\item " [ math define proof of lemma equivalence class is subset as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var big set end metavar indeed metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set infer lemma separation2formula modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 zermelo pair zermelo pair metavar var s end metavar comma metavar var s end metavar end pair comma zermelo pair metavar var s end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude metavar var s end metavar zermelo in metavar var big set end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var big set end metavar indeed metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set infer metavar var s end metavar zermelo in metavar var big set end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply metavar var s end metavar zermelo in metavar var big set end metavar end quote state proof state cache var c end expand end define end math ] "
\item Kommet s\aa{} vidt kan vi bevise, at " [ math metavar var big set end metavar end math ] " er en delm\ae{}ngde af " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] ":
\item " [ math define statement of lemma bs subset union(bs/r) as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end define end math ] "
\item Beviset for " [ math lemma bs subset union(bs/r) end math ] " falder i tre dele:
%
\begin{enumerate}
\item Lad " [ math metavar var s end metavar end math ] " v\ae{}re et vilk\aa{}rligt medlem af " [ math metavar var big set end metavar end math ] ". Ud fra " [ math lemma auto member end math ] " har vi, at " [ math metavar var s end metavar end math ] " tilh\o{}rer sin egen \ae{}kvivalensklasse " [ math the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set end math ] " (linie 6--7).
\item Videre har vi, at denne \ae{}kvivalensklasse tilh\o{}rer \ae{}kvivalenssystemet \\
\mbox{" [ math the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end math ] "}
(linie 8--13).
\item Ud fra punkt 1 og 2 og definitionen af ``foreningsm\ae{}ngde'' f\aa{}r vi, at " [ math metavar var s end metavar end math ] " tilh\o{}rer \ae{}kvivalenssystemets foreningsm\ae{}ngde " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] ", QED \\ \mbox{(linie 14--16)}.
\end{enumerate}
%
Her er selve beviset:
\item " [ math define proof of lemma bs subset union(bs/r) as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer lemma auto member modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma equivalence class is subset conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in metavar var big set end metavar cut lemma subset in power set modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in metavar var big set end metavar conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in power metavar var big set end metavar end power cut lemma =reflexivity conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma join conjuncts modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut 1rule exist intro at existential var var a end var at metavar var s end metavar modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma formula2separation modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in power metavar var big set end metavar end power modus ponens not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut 1rule exist intro at existential var var j end var at the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude metavar var s end metavar zermelo in existential var var j end var cut 1rule exist intro at existential var var j end var at the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut lemma formula2union modus ponens metavar var s end metavar zermelo in existential var var j end var modus ponens existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent I beviset bruger vi for f\o{}rste gang slutningsreglen " [ math 1rule exist intro end math ] ". Vi anvender her konstruktionen " [ bracket var x at var y end bracket ] " til at fort\ae{}lle bevischeckeren, hvordan metavariablene " [ math metavar var x end metavar end math ] " og " [ math metavar var t end metavar end math ] " i " [ math 1rule exist intro end math ] " skal instantieres. Dette kan bevischeckeren ikke selv finde frem til, da b\aa{}de ``" [ math metavar var x end metavar end math ] "'' og ``" [ math metavar var t end metavar end math ] "''
kun optr\ae{}der i reglens sidebetingelse. Det er med andre ord det samme problem som ved separationsaksiomet i afsnit \ref{sec:separation}, som vi her har l\o{}st p\aa{} en anden m\aa{}de.
\subsubsection{Den anden halvdel} \label{sec:union2}
Vi viser nu det modsatte resultat af " [ math lemma bs subset union(bs/r) end math ] ": " [ math metavar var big set end metavar end math ] " er en delm\ae{}ngde af " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] ":
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\item " [ math define statement of lemma union(bs/r) subset bs as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply metavar var s end metavar zermelo in metavar var big set end metavar end define end math ] "
\item Beviset for " [ math lemma union(bs/r) subset bs end math ] " kan opdeles i fire dele:
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\begin{enumerate}
\item Lad " [ math metavar var s end metavar end math ] " v\ae{}re et medlem af " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] ". " [ bracket metavar var s end metavar end bracket ] " m\aa{} tilh\o{}re en \ae{}kvivalensklasse, som vi kalder for " [ math existential var var j end var end math ] " (linie 5--7).
\item " [ bracket existential var var j end var end bracket ] " tilh\o{}rer \ae{}kvivalenssystemet " [ math the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end math ] ", som igen er en delm\ae{}ngde af potensm\ae{}ngden " [ math power metavar var big set end metavar end power end math ] ". Derfor m\aa{} " [ math existential var var j end var end math ] " tilh\o{}re " [ math power metavar var big set end metavar end power end math ] " (linie 8--10).
\item Dette medf\o{}rer igen, at " [ math existential var var j end var end math ] " er en delm\ae{}ngde af " [ math metavar var big set end metavar end math ] " (linie 11).
\item Ud fra punkt 1 og 3 f\aa{}r vi, at " [ math metavar var s end metavar end math ] " tilh\o{}rer " [ math metavar var big set end metavar end math ] ", QED (linie 12--14).
\end{enumerate}
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Her er selve beviset:
\item " [ math define proof of lemma union(bs/r) subset bs as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union infer lemma union2formula modus ponens metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union conclude not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut prop lemma first conjunct modus ponens not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude metavar var s end metavar zermelo in existential var var j end var cut prop lemma second conjunct modus ponens not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut lemma separation2formula modus ponens existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude not0 existential var var j end var zermelo in power metavar var big set end metavar end power imply not0 not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is existential var var j end var cut prop lemma first conjunct modus ponens not0 existential var var j end var zermelo in power metavar var big set end metavar end power imply not0 not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is existential var var j end var conclude existential var var j end var zermelo in power metavar var big set end metavar end power cut lemma power set is subset-switch modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens existential var var j end var zermelo in power metavar var big set end metavar end power conclude object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar cut 1rule gen modus ponens object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar conclude for all objects object var var s end var indeed object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar cut lemma power set is subset0-switch modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote conclude for all objects object var var s end var indeed object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in existential var var j end var imply metavar var s end metavar zermelo in metavar var big set end metavar cut prop lemma mp2 modus ponens for all objects object var var s end var indeed object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in existential var var j end var imply metavar var s end metavar zermelo in metavar var big set end metavar modus ponens for all objects object var var s end var indeed object var var s end var zermelo in existential var var j end var imply object var var s end var zermelo in metavar var big set end metavar modus ponens metavar var s end metavar zermelo in existential var var j end var conclude metavar var s end metavar zermelo in metavar var big set end metavar cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union infer metavar var s end metavar zermelo in metavar var big set end metavar conclude quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply metavar var s end metavar zermelo in metavar var big set end metavar end quote state proof state cache var c end expand end define end math ] "
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\subsubsection{De to halvdele s\ae{}ttes sammen} \label{sec:union3}
Nu hvor vi har vist, at " [ math metavar var big set end metavar end math ] " og " [ math union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end math ] " er hinandens delm\ae{}ngder, er det let at vise, at de er lig med hinanden:
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\item " [ math define statement of lemma union(bs/r) is bs as system zf infer all metavar var r end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union zermelo is metavar var big set end metavar end define end math ] "
\item " [ math define proof of lemma union(bs/r) is bs as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer lemma bs subset union(bs/r) modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in metavar var big set end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union cut 1rule mp modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in metavar var big set end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in metavar var big set end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union cut lemma union(bs/r) subset bs modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply object var var s end var zermelo in metavar var big set end metavar cut 1rule mp modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply object var var s end var zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply object var var s end var zermelo in metavar var big set end metavar cut lemma set equality suff condition modus ponens object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union imply object var var s end var zermelo in metavar var big set end metavar modus ponens object var var s end var zermelo in metavar var big set end metavar imply object var var s end var zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union conclude union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union zermelo is metavar var big set end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Det sidste bevis} \label{sec:sidste}
Rapportens sidste bevis vedr\o{}rer naturligvis hovedresultatet:
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\item " [ math define statement of theorem eq-system is partition as system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var end define end math ] "
\item I beviset sammens\ae{}tter vi resultaterne fra underafsnit \ref{sec:nummer1} (linie 2--4), underafsnit \ref{sec:nummer2} (linie 5--8) og underafsnit \ref{sec:nummer3} (linie 9):
\item " [ math define proof of theorem eq-system is partition as lambda var c dot lambda var x dot proof expand quote system zf infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var infer lemma eq-system not empty modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var conclude object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var x end var zermelo is zermelo empty set cut 1rule deduction modus ponens object var var x end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var x end var zermelo is zermelo empty set conclude object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set cut 1rule gen modus ponens object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set conclude for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set cut lemma all disjoint-imply conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var imply object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set cut 1rule mp modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var imply object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var conclude object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set cut 1rule gen modus ponens object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set conclude for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set cut 1rule gen modus ponens for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set cut lemma union(bs/r) is bs modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in object var var r end var imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in object var var big set end var imply object var var t end var zermelo in object var var big set end var imply object var var u end var zermelo in object var var big set end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in object var var r end var imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in object var var r end var conclude union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var cut prop lemma join conjuncts modus ponens for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set conclude not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set cut prop lemma join conjuncts modus ponens not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set modus ponens union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var cut 1rule repetition modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var conclude not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is zermelo empty set imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply object var var t end var zermelo in the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set imply not0 object var var s end var zermelo is object var var t end var imply the set of ph in union zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var t end var comma object var var t end var end pair end pair end union such that not0 placeholder-var var c end var zermelo in object var var s end var imply not0 placeholder-var var c end var zermelo in object var var t end var end set zermelo is zermelo empty set imply not0 union the set of ph in power object var var big set end var end power such that not0 existential var var t end var zermelo in object var var big set end var imply not0 the set of ph in object var var big set end var such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in object var var r end var end set zermelo is placeholder-var var b end var end set end union zermelo is object var var big set end var end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent L\ae{}g m\ae{}rke til, at vi i linie 2 instantierer meta-variablen " [ math metavar var s end metavar end math ] " i " [ math lemma eq-system not empty end math ] " til objektvariablen " [ math object var var x end var end math ] ", selvom det egentlig er " [ math object var var s end var end math ] ", vi gerne ville have fat i. Dette skyldes, at " [ math lemma eq-system not empty end math ] " indeholder sidebetingelsen " [ bracket quote object var var s end var end quote avoid zero quote metavar var s end metavar end quote end bracket ] " (jvf.\ afsnit \ref{sec:nummer1}). Vi l\o{}ser problemet i linie 3, hvor deduktionsreglen skifter " [ math object var var x end var end math ] " ud med " [ math object var var s end var end math ] ". Deduktionsreglen tillader kun s\aa{}danne variabelskift, n\aa{}r der ikke er nogen metavariable i n\ae{}rheden; det er derfor, at vi ikke tidligere har brugt reglen p\aa{} denne m\aa{}de.
\newpage
\section{Konklusion} \label{sec:kon}
Vi har ikke haft 100\% succes med at bevise hovedresultatet; der m\aa{}tte et ``snyde-aksiom'' til hj\ae{}lp. Problemet er, at systemet kr\ae{}ver et stort antal sidebetingelser for at sikre, at objekt- og metavariable undg\aa{}r hinanden. Et sp\o{}rgsm\aa{}l er, om vi kunne have undg\aa{}et dette problem ved at h\aa{}ndtere aksiomerne og definitionerne fra afsnit \ref{sec:aksiomsystem} og \ref{sec:makro} anderledes. Her er en kort diskussion af et par alternative strategier:
Det er n\ae{}ppe holdbart kun at arbejde med objektvariable. Deduktionsreglen kan gennemf\o{}re variabelskift som f.eks.\ ``" [ math object var var y end var end math ] " i stedet for " [ math object var var x end var end math ] "'', men ikke instantieringer som f.eks.\ ``" [ math zermelo pair object var var y end var comma object var var z end var end pair end math ] " i stedet for " [ math object var var x end var end math ] "''. (Jvf.\ formuleringen af " [ math lemma same intersection end math ] " i afsnit \ref{sec:sameinter}, hvor vi m\aa{}tte instantiere objektvariablene ``p\aa{} forh\aa{}nd''). Vi s\aa{} ogs\aa{} i afsnit \ref{sec:konvertering}, at det er sv\ae{}rt at skifte fra inferens til implikation med objektvariable; deduktionsreglen kan ikke anvendes.
Det andet ekstrem er kun at arbejde med metavariable og -kvantorer. Man kunne da f.eks.\ formulere den ene halvdel af " [ math axiom extensionality end math ] " som
\display{" [ bracket all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var s end metavar indeed not0 metavar var s end metavar zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in metavar var y end metavar imply not0 metavar var s end metavar zermelo in metavar var y end metavar imply metavar var s end metavar zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is metavar var y end metavar end bracket ] ".}
\noindent Logiwebs bevischecker er imidlertid ikke gearet til denne l\o{}sning, og det virker heller ikke som om, at vi p\aa{} denne m\aa{}de l\o{}ser det egentlige problem: F.eks.\ giver det noget sludder, hvis vi instantierer ``" [ math metavar var s end metavar end math ] "'' til ``" [ math metavar var x end metavar end math ] "'' i det ovenst\aa{}ende. Den samme kritik kan rettes mod forslaget ``lad objektkvantorerne binde metavariable i stedet for objektvariable''.
Den bedste l\o{}sning m\aa{} derfor v\ae{}re at lade bevischeckeren f\aa{} ansvaret for at administrere sidebetingelserne, s\aa{}ledes at brugeren kun l\ae{}gger m\ae{}rke til dem, hvis han overtr\ae{}der dem. Jeg har imidlertid ikke noget overblik over, hvor kr\ae{}vende det er at implementere denne l\o{}sning.
Min anden hovedkonklusion er, at deduktionsreglen skal suppleres med A4 fra \cite{kn:mendel}, hvis eksistenskvantorer skal h\aa{}ndteres tilfredsstillende. Min l\o{}sning med eksistens-variable har fungeret i denne rapport, men det er ikke nogen sikker metode. Der er f.eks.\ intet, som forhindrer brugeren i at introducere allerede brugte eksistens-variable og dermed opn\aa{} systemets godkendelse af et ukorrekt bevis. Det er et \aa{}bent sp\o{}rgsm\aa{}l, hvor brugervenlig kombinationen ``" [ math 1rule deduction end math ] " og A4'' er; den interesserede l\ae{}ser skal v\ae{}re velkommen til at g\o{}re sine egne erfaringer.
\newpage
\section*{}
\addcontentsline{toc}{section}{Litteratur}
\bibliography{/net/urd/home/disk15/eriksen/.logiweb/equivalence-relations/page}
\appendix
\section{Oversigt over variabelnavne} \label{sec:variable}
\begin{itemize}
\item De fire vigtigste objektvariable er: " [ bracket object var var s end var end bracket ] ", " [ bracket object var var t end var end bracket ] ", " [ bracket object var var u end var end bracket ] " og " [ bracket object var var big set end var end bracket ] ".
\item Eksistens-variablen " [ bracket existential var var j end var end bracket ] " bruges i aksiomet
" [ bracket axiom union definition end bracket ] " fra afsnit \ref{sec:axiomzf}.
\item Ekistens-variablen " [ bracket existential var var t end var end bracket ] " bruges i definitionen
af " [ bracket the set of ph in power var big set end power such that not0 existential var var t end var zermelo in var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in var r end set zermelo is placeholder-var var b end var end set end bracket ] " fra afsnit
\ref{sec:eqclass}.
\item Eksistens-variablene " [ bracket existential var var a end var end bracket ] " og " [ bracket existential var var b end var end bracket ] " bruges i beviser, n\aa{}r " [ bracket existential var var j end var end bracket ] " eller " [ bracket existential var var t end var end bracket ] " ikke er p\aa{}kr\ae{}vede.
\item Pladsholder-variablen " [ bracket placeholder-var var a end var end bracket ] " bruges i definitionen af " [ bracket the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma var x end pair end pair zermelo in var r end set end bracket ] " fra afsnit \ref{sec:eqclass}.
\item Pladsholder-variablen " [ bracket placeholder-var var b end var end bracket ] " bruges i definitionen
af " [ bracket the set of ph in power var big set end power such that not0 existential var var t end var zermelo in var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in var r end set zermelo is placeholder-var var b end var end set end bracket ] " fra afsnit
\ref{sec:eqclass}.
\item Pladsholder-variablen " [ bracket placeholder-var var c end var end bracket ] " bruges i definitionen af " [ bracket the set of ph in union zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var y comma var y end pair end pair end union such that not0 placeholder-var var c end var zermelo in var x imply not0 placeholder-var var c end var zermelo in var y end set end bracket ] " fra afsnit \ref{sec:makrobin}.
\item Metavariablene " [ bracket metavar var a end metavar end bracket ] ", " [ bracket metavar var b end metavar end bracket ] " $\cdots$ " [ bracket metavar var f end metavar end bracket ] " varierer over formler.
\item Metavariabelene " [ bracket metavar var r end metavar end bracket ] ", " [ bracket metavar var s end metavar end bracket ] " $\cdots$ " [ bracket metavar var z end metavar end bracket ] " samt " [ bracket metavar var big set end metavar end bracket ] " varierer over termer. (" [ bracket metavar var r end metavar end bracket ] " st\aa{}r altid for en relation).
\item Metavariablen " [ bracket metavar var p end metavar end bracket ] " varierer over pladsholder-variable.
\item Endelig kan de ``r\aa{}'' variable " [ bracket var a end bracket ] ", " [ bracket var b end bracket ] " $\cdots$ " [ bracket var z end bracket ] " samt " [ bracket var big set end bracket ] " variere over hvad som helst; disse variable er ikke tilknyttet nogen semantik.
\end{itemize}
\section{Det samlede aksiomsystem} \label{sec:helezf}
\begin{list}{}{
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of 1rule mp as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule mp as rule tactic end define end math ] "
\item " [ math define statement of 1rule gen as system zf infer all metavar var x end metavar indeed all metavar var a end metavar indeed metavar var a end metavar infer for all objects metavar var x end metavar indeed metavar var a end metavar end define , define proof of 1rule gen as rule tactic end define end math ] "
\item " [ math define statement of 1rule repetition as system zf infer all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end define , define proof of 1rule repetition as rule tactic end define end math ] "
\item " [ math define statement of 1rule ad absurdum as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule ad absurdum as rule tactic end define end math ] "
\item " [ math define statement of 1rule deduction as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot 1deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end 1deduction endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule deduction as rule tactic end define end math ] "
\item " [ math define statement of 1rule exist intro as system zf infer all metavar var x end metavar indeed all metavar var t end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed exist-sub0 quote metavar var a end metavar end quote is quote metavar var b end metavar end quote where quote metavar var x end metavar end quote is quote metavar var t end metavar end quote end sub endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule exist intro as rule tactic end define end math ] "
\item " [ math define statement of axiom extensionality as system zf infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var zermelo in metavar var x end metavar imply object var var s end var zermelo in metavar var y end metavar imply not0 object var var s end var zermelo in metavar var y end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end define , define proof of axiom extensionality as rule tactic end define end math ] "
\item " [ math define statement of axiom empty set as system zf infer all metavar var s end metavar indeed not0 metavar var s end metavar zermelo in zermelo empty set end define , define proof of axiom empty set as rule tactic end define end math ] "
\item " [ math define statement of axiom pair definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar zermelo in zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end define , define proof of axiom pair definition as rule tactic end define end math ] "
\item " [ math define statement of axiom union definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in union metavar var x end metavar end union imply not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply not0 not0 metavar var s end metavar zermelo in existential var var j end var imply not0 existential var var j end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in union metavar var x end metavar end union end define , define proof of axiom union definition as rule tactic end define end math ] "
\item " [ math define statement of axiom power definition as system zf infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar zermelo in power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var s end metavar imply object var var s end var zermelo in metavar var x end metavar imply metavar var s end metavar zermelo in power metavar var x end metavar end power end define , define proof of axiom power definition as rule tactic end define end math ] "
\item " [ math define statement of axiom separation definition as system zf infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var z end metavar indeed metavar var p end metavar is placeholder-var and ph-sub0 quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var p end metavar end quote is quote metavar var z end metavar end quote end sub endorse not0 metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var z end metavar zermelo in metavar var x end metavar imply not0 metavar var b end metavar imply metavar var z end metavar zermelo in the set of ph in metavar var x end metavar such that metavar var a end metavar end set end define , define proof of axiom separation definition as rule tactic end define end math ] "
\item " [ math define statement of cheating axiom all disjoint as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var x end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer metavar var y end metavar zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set infer not0 metavar var x end metavar zermelo is metavar var y end metavar infer the set of ph in union zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var y end metavar end pair end pair end union such that not0 placeholder-var var c end var zermelo in metavar var x end metavar imply not0 placeholder-var var c end var zermelo in metavar var y end metavar end set zermelo is zermelo empty set end define , define proof of cheating axiom all disjoint as rule tactic end define end math ] "
\end{list}
\newpage
\section{Deduktionsreglen} \label{sec:deduktion}
Dette bilag pr\ae{}senterer d\'{e}n version af deduktionsreglen fra \cite{kn:check}, som jeg har brugt af. Underafsnit \ref{sec:motivering} forklarer, hvorfor jeg har \ae{}ndret p\aa{} den oprindelige regel, og underafsnit \ref{sec:kode} indeholder selve den \ae{}ndrede kode (som er skrevet i L).
\subsection{Motivering} \label{sec:motivering}
I beviset for " [ math lemma member not empty0 end math ] " i afsnit \ref{sec:membernotempty} konkluderer vi \\
" [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar imply metavar var x end metavar zermelo is zermelo empty set imply metavar var s end metavar zermelo in zermelo empty set end bracket ] " ud fra pr\ae{}missen \\
" [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote endorse metavar var s end metavar zermelo in metavar var x end metavar infer metavar var x end metavar zermelo is zermelo empty set infer metavar var s end metavar zermelo in zermelo empty set end bracket ] " ved hj\ae{}lp af deduktionsreglen; det er et klassisk skift fra inferens til implikation. Denne slutning kan imidlertid ikke gennemf\o{}res med deduktionsreglen fra \cite{kn:check}, fordi denne regel fjerner alle sidebetingelser fra konklusionen, f\o{}r pr\ae{}mis og konklusion sammenlignes. Da pr\ae{}missen stadigv\ae{}k begynder med sidebetingelsen " [ bracket quote object var var s end var end quote avoid zero quote metavar var x end metavar end quote end bracket ] ", matcher pr\ae{}mis og konklusion ikke hinanden, og slutningen fra pr\ae{}mis til konklusion forkastes. Deduktionsreglen fra dette bilag tillader derimod, at pr\ae{}mis og konklusion begynder med et antal identiske sidebetingelser. Dermed kan vi uden problemer gennemf\o{}re slutninger som den ovenst\aa{}ende, hvor pr\ae{}missen indeholder sidebetingelser.
\subsection{Kode} \label{sec:kode}
Funktionen " [ bracket lambda var x dot 1deduction zero quote var p end quote conclude quote var c end quote end 1deduction end bracket ] " er en kopi af " [ bracket lambda var x dot deduction zero quote var p end quote conclude quote var c end quote end deduction end bracket ] " fra \cite{kn:check}:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define macro of 1deduction var p conclude var c end 1deduction as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define 1deduction var p conclude var c end 1deduction as lambda var x dot 1deduction zero quote var p end quote conclude quote var c end quote end 1deduction end define end quote end define end define end math ] "
%(*** AENDRING BEGYNDER ***)
\item Jeg har \ae{}ndret funktionen " [ bracket deduction zero var p conclude var c end deduction end bracket ] ", s\aa{} den kalder " [ bracket 1deduction side 1deduction seven var p end 1deduction conclude var c condition true end 1deduction end bracket ] " i stedet for " [ bracket deduction one deduction seven var p end deduction conclude var c condition true end deduction end bracket ] ":
\item " [ math define value of 1deduction zero var p conclude var c end 1deduction as var c tagged guard tagged if 1deduction eight var p bound true end 1deduction then 1deduction side 1deduction seven var p end 1deduction conclude var c condition true end 1deduction else false end if end define end math ] "
\item Funktionen " [ bracket 1deduction side var p conclude var c condition var s end 1deduction end bracket ] " giver straks kontrollen videre til " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] " --- medmindre " [ math var p end math ] " og " [ math var c end math ] " begynder med et antal identiske sidebetingelser. I s\aa{} fald flyttes disse sidebetingelser fra " [ math var p end math ] " og " [ math var c end math ] " over til listen " [ math var s end math ] ", f\o{}r kontrollen g\aa{}r videre til " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] ":
\item " [ math define value of 1deduction side var p conclude var c condition var s end 1deduction as tagged if var p term root equal quote var x endorse var y end quote then var c term root equal quote var x endorse var y end quote and var p first term equal var c first and 1deduction side var p second conclude var c second condition var c first pair var s end 1deduction else 1deduction one var p conclude var c condition var s end 1deduction end if end define end math ] "
%(*** AENDRING SLUTTER ***)
\item Fra og med " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] " er koden kopieret fra appendikset til \cite{kn:check}:
\item
" [ math define value of 1deduction one var p conclude var c condition var s end 1deduction as tagged if var c term root equal quote var x endorse var y end quote then 1deduction one var p conclude var c second condition var c first pair var s end 1deduction else 1deduction two var p conclude var c condition var s end 1deduction end if end define end math ] "
\item " [ math define value of 1deduction two var p conclude var c condition var s end 1deduction as var s tagged guard var p term root equal quote var x infer var y end quote and var c term root equal quote var x imply var y end quote select 1deduction three var p first conclude var c first condition var s bound true end 1deduction and 1deduction two var p second conclude var c second condition var s end 1deduction else 1deduction four var p conclude var c condition var s bound 1deduction six var p conclude var c exception true bound true end 1deduction end 1deduction end select end define end math ] "
\item " [ math define value of 1deduction three var p conclude var c condition var s bound var b end 1deduction as tagged if not var c term root equal quote for all objects var x indeed var y end quote then 1deduction four var p conclude var c condition var s bound var b end 1deduction else tagged if var p term root equal quote for all objects var x indeed var y end quote and var p first term equal var c first then 1deduction four var p conclude var c condition var s bound var b end 1deduction else 1deduction three var p conclude var c second condition var s bound var c first pair var c first pair var b end 1deduction end if end if end define end math ] "
\item " [ math define value of 1deduction four var p conclude var c condition var s bound var b end 1deduction as var s tagged guard var b tagged guard tagged if var p term root equal quote object var var x end var end quote then lookup var p stack var b default true end lookup term equal var c else tagged if not var p term root equal var c then false else tagged if var p term root equal quote for all objects var x indeed var y end quote then var p first term equal var c first and 1deduction four var p second conclude var c second condition var s bound var p first pair var p first pair var b end 1deduction else tagged if not var p term root equal quote metavar var x end metavar end quote then 1deduction four star var p tail conclude var c tail condition var s bound var b end 1deduction else var p first term equal var c first and 1deduction five var p condition var s bound var b end 1deduction end if end if end if end if end define end math ] "
\item " [ math define value of 1deduction four star var p conclude var c condition var s bound var b end 1deduction as var c tagged guard var s tagged guard var b tagged guard tagged if var p then true else 1deduction four var p head conclude var c head condition var s bound var b end 1deduction and 1deduction four star var p tail conclude var c tail condition var s bound var b end 1deduction end if end define end math ] "
\item " [ math define value of 1deduction five var p condition var s bound var b end 1deduction as var p tagged guard var s tagged guard tagged if var b then true else quote quote var x end quote avoid zero quote var y end quote end quote head pair quote quote x end quote end quote head pair var b head head pair true pair quote quote var x end quote end quote head pair var p pair true pair true term in var s and 1deduction five var p condition var s bound var b tail end 1deduction end if end define end math ] "
\item " [ math define value of 1deduction six var p conclude var c exception var e bound var b end 1deduction as var p tagged guard var c tagged guard var b tagged guard var e tagged guard tagged if var p term root equal quote object var var x end var end quote then var p term in var e select var b else var p pair var c pair var b end select else tagged if not var p term root equal var c then true else tagged if var p term root equal quote metavar var a end metavar end quote then var b else tagged if var p term root equal quote for all objects var x indeed var y end quote then 1deduction six var p second conclude var c second exception var c first pair var e bound var b end 1deduction else 1deduction six star var p tail conclude var c tail exception var e bound var b end 1deduction end if end if end if end if end define end math ] "
\item " [ math define value of 1deduction six star var p conclude var c exception var e bound var b end 1deduction as var p tagged guard var c tagged guard var b tagged guard var e tagged guard tagged if var p then var b else 1deduction six star var p tail conclude var c tail exception var e bound 1deduction six var p head conclude var c head exception var e bound var b end 1deduction end 1deduction end if end define end math ] "
\item " [ math define value of 1deduction seven var p end 1deduction as var p term root equal quote all var x indeed var y end quote select 1deduction seven var p second end 1deduction else var p end select end define end math ] "
\item " [ math define value of 1deduction eight var p bound var b end 1deduction as tagged if var p term root equal quote all var x indeed var y end quote then 1deduction eight var p second bound var p first pair var b end 1deduction else tagged if var p term root equal quote metavar var a end metavar end quote then var p term in var b else 1deduction eight star var p tail bound var b end 1deduction end if end if end define end math ] "
\item " [ math define value of 1deduction eight star var p bound var b end 1deduction as var b tagged guard tagged if var p then true else tagged if 1deduction eight var p head bound var b end 1deduction then 1deduction eight star var p tail bound var b end 1deduction else false end if end if end define end math ] "
\end{list}
\section{Pyk definitioner} \label{sec:pyk}
\begin{flushleft}
" [ math define pyk of cdots as text "cdots" end text end define linebreak define pyk of object-var as text "object-var" end text end define linebreak define pyk of ex-var as text "ex-var" end text end define linebreak define pyk of ph-var as text "ph-var" end text end define linebreak define pyk of vaerdi as text "vaerdi" end text end define linebreak define pyk of variabel as text "variabel" end text end define linebreak define pyk of op x end op as text "op "! end op" end text end define linebreak define pyk of op2 x comma x end op2 as text "op2 "! comma "! end op2" end text end define linebreak define pyk of define-equal x comma x end equal as text "define-equal "! comma "! end equal" end text end define linebreak define pyk of contains-empty x end empty as text "contains-empty "! end empty" end text end define linebreak define pyk of 1deduction x conclude x end 1deduction as text "1deduction "! conclude "! end 1deduction" end text end define linebreak define pyk of 1deduction zero x conclude x end 1deduction as text "1deduction zero "! conclude "! end 1deduction" end text end define linebreak define pyk of 1deduction side x conclude x condition x end 1deduction as text "1deduction side "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction one x conclude x condition x end 1deduction as text "1deduction one "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction two x conclude x condition x end 1deduction as text "1deduction two "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction three x conclude x condition x bound x end 1deduction as text "1deduction three "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction four x conclude x condition x bound x end 1deduction as text "1deduction four "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction four star x conclude x condition x bound x end 1deduction as text "1deduction four star "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction five x condition x bound x end 1deduction as text "1deduction five "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction six x conclude x exception x bound x end 1deduction as text "1deduction six "! conclude "! exception "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction six star x conclude x exception x bound x end 1deduction as text "1deduction six star "! conclude "! exception "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction seven x end 1deduction as text "1deduction seven "! end 1deduction" end text end define linebreak define pyk of 1deduction eight x bound x end 1deduction as text "1deduction eight "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction eight star x bound x end 1deduction as text "1deduction eight star "! bound "! end 1deduction" end text end define linebreak define pyk of ex1 as text "ex1" end text end define linebreak define pyk of ex2 as text "ex2" end text end define linebreak define pyk of ex10 as text "ex10" end text end define linebreak define pyk of ex20 as text "ex20" end text end define linebreak define pyk of existential var x end var as text "existential var "! end var" end text end define linebreak define pyk of x is existential var as text ""! is existential var" end text end define linebreak define pyk of exist-sub x is x where x is x end sub as text "exist-sub "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub0 x is x where x is x end sub as text "exist-sub0 "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub1 x is x where x is x end sub as text "exist-sub1 "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub* x is x where x is x end sub as text "exist-sub* "! is "! where "! is "! end sub" end text end define linebreak define pyk of placeholder-var1 as text "placeholder-var1" end text end define linebreak define pyk of placeholder-var2 as text "placeholder-var2" end text end define linebreak define pyk of placeholder-var3 as text "placeholder-var3" end text end define linebreak define pyk of placeholder-var x end var as text "placeholder-var "! end var" end text end define linebreak define pyk of x is placeholder-var as text ""! is placeholder-var" end text end define linebreak define pyk of ph-sub x is x where x is x end sub as text "ph-sub "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub0 x is x where x is x end sub as text "ph-sub0 "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub1 x is x where x is x end sub as text "ph-sub1 "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub* x is x where x is x end sub as text "ph-sub* "! is "! where "! is "! end sub" end text end define linebreak define pyk of var big set as text "var big set" end text end define linebreak define pyk of object big set as text "object big set" end text end define linebreak define pyk of meta big set as text "meta big set" end text end define linebreak define pyk of zermelo empty set as text "zermelo empty set" end text end define linebreak define pyk of system zf as text "system zf" end text end define linebreak define pyk of 1rule mp as text "1rule mp" end text end define linebreak define pyk of 1rule gen as text "1rule gen" end text end define linebreak define pyk of 1rule repetition as text "1rule repetition" end text end define linebreak define pyk of 1rule ad absurdum as text "1rule ad absurdum" end text end define linebreak define pyk of 1rule deduction as text "1rule deduction" end text end define linebreak define pyk of 1rule exist intro as text "1rule exist intro" end text end define linebreak define pyk of axiom extensionality as text "axiom extensionality" end text end define linebreak define pyk of axiom empty set as text "axiom empty set" end text end define linebreak define pyk of axiom pair definition as text "axiom pair definition" end text end define linebreak define pyk of axiom union definition as text "axiom union definition" end text end define linebreak define pyk of axiom power definition as text "axiom power definition" end text end define linebreak define pyk of axiom separation definition as text "axiom separation definition" end text end define linebreak define pyk of cheating axiom all disjoint as text "cheating axiom all disjoint" end text end define linebreak define pyk of prop lemma add double neg as text "prop lemma add double neg" end text end define linebreak define pyk of prop lemma remove double neg as text "prop lemma remove double neg" end text end define linebreak define pyk of prop lemma and commutativity as text "prop lemma and commutativity" end text end define linebreak define pyk of prop lemma auto imply as text "prop lemma auto imply" end text end define linebreak define pyk of prop lemma contrapositive as text "prop lemma contrapositive" end text end define linebreak define pyk of prop lemma first conjunct as text "prop lemma first conjunct" end text end define linebreak define pyk of prop lemma second conjunct as text "prop lemma second conjunct" end text end define linebreak define pyk of prop lemma from contradiction as text "prop lemma from contradiction" end text end define linebreak define pyk of prop lemma from disjuncts as text "prop lemma from disjuncts" end text end define linebreak define pyk of prop lemma iff commutativity as text "prop lemma iff commutativity" end text end define linebreak define pyk of prop lemma iff first as text "prop lemma iff first" end text end define linebreak define pyk of prop lemma iff second as text "prop lemma iff second" end text end define linebreak define pyk of prop lemma imply transitivity as text "prop lemma imply transitivity" end text end define linebreak define pyk of prop lemma join conjuncts as text "prop lemma join conjuncts" end text end define linebreak define pyk of prop lemma mp2 as text "prop lemma mp2" end text end define linebreak define pyk of prop lemma mp3 as text "prop lemma mp3" end text end define linebreak define pyk of prop lemma mp4 as text "prop lemma mp4" end text end define linebreak define pyk of prop lemma mp5 as text "prop lemma mp5" end text end define linebreak define pyk of prop lemma mt as text "prop lemma mt" end text end define linebreak define pyk of prop lemma negative mt as text "prop lemma negative mt" end text end define linebreak define pyk of prop lemma technicality as text "prop lemma technicality" end text end define linebreak define pyk of prop lemma weakening as text "prop lemma weakening" end text end define linebreak define pyk of prop lemma weaken or first as text "prop lemma weaken or first" end text end define linebreak define pyk of prop lemma weaken or second as text "prop lemma weaken or second" end text end define linebreak define pyk of lemma formula2pair as text "lemma formula2pair" end text end define linebreak define pyk of lemma pair2formula as text "lemma pair2formula" end text end define linebreak define pyk of lemma formula2union as text "lemma formula2union" end text end define linebreak define pyk of lemma union2formula as text "lemma union2formula" end text end define linebreak define pyk of lemma formula2separation as text "lemma formula2separation" end text end define linebreak define pyk of lemma separation2formula as text "lemma separation2formula" end text end define linebreak define pyk of lemma subset in power set as text "lemma subset in power set" end text end define linebreak define pyk of lemma power set is subset0 as text "lemma power set is subset0" end text end define linebreak define pyk of lemma power set is subset as text "lemma power set is subset" end text end define linebreak define pyk of lemma power set is subset0-switch as text "lemma power set is subset0-switch" end text end define linebreak define pyk of lemma power set is subset-switch as text "lemma power set is subset-switch" end text end define linebreak define pyk of lemma set equality suff condition as text "lemma set equality suff condition" end text end define linebreak define pyk of lemma set equality suff condition(t)0 as text "lemma set equality suff condition(t)0" end text end define linebreak define pyk of lemma set equality suff condition(t) as text "lemma set equality suff condition(t)" end text end define linebreak define pyk of lemma set equality skip quantifier as text "lemma set equality skip quantifier" end text end define linebreak define pyk of lemma set equality nec condition as text "lemma set equality nec condition" end text end define linebreak define pyk of lemma reflexivity0 as text "lemma reflexivity0" end text end define linebreak define pyk of lemma reflexivity as text "lemma reflexivity" end text end define linebreak define pyk of lemma symmetry0 as text "lemma symmetry0" end text end define linebreak define pyk of lemma symmetry as text "lemma symmetry" end text end define linebreak define pyk of lemma transitivity0 as text "lemma transitivity0" end text end define linebreak define pyk of lemma transitivity as text "lemma transitivity" end text end define linebreak define pyk of lemma er is reflexive as text "lemma er is reflexive" end text end define linebreak define pyk of lemma er is symmetric as text "lemma er is symmetric" end text end define linebreak define pyk of lemma er is transitive as text "lemma er is transitive" end text end define linebreak define pyk of lemma empty set is subset as text "lemma empty set is subset" end text end define linebreak define pyk of lemma member not empty0 as text "lemma member not empty0" end text end define linebreak define pyk of lemma member not empty as text "lemma member not empty" end text end define linebreak define pyk of lemma unique empty set0 as text "lemma unique empty set0" end text end define linebreak define pyk of lemma unique empty set as text "lemma unique empty set" end text end define linebreak define pyk of lemma =reflexivity as text "lemma =reflexivity" end text end define linebreak define pyk of lemma =symmetry as text "lemma =symmetry" end text end define linebreak define pyk of lemma =transitivity0 as text "lemma =transitivity0" end text end define linebreak define pyk of lemma =transitivity as text "lemma =transitivity" end text end define linebreak define pyk of lemma transfer ~is0 as text "lemma transfer ~is0" end text end define linebreak define pyk of lemma transfer ~is as text "lemma transfer ~is" end text end define linebreak define pyk of lemma pair subset0 as text "lemma pair subset0" end text end define linebreak define pyk of lemma pair subset1 as text "lemma pair subset1" end text end define linebreak define pyk of lemma pair subset as text "lemma pair subset" end text end define linebreak define pyk of lemma same pair as text "lemma same pair" end text end define linebreak define pyk of lemma same singleton as text "lemma same singleton" end text end define linebreak define pyk of lemma union subset as text "lemma union subset" end text end define linebreak define pyk of lemma same union as text "lemma same union" end text end define linebreak define pyk of lemma separation subset as text "lemma separation subset" end text end define linebreak define pyk of lemma same separation as text "lemma same separation" end text end define linebreak define pyk of lemma same binary union as text "lemma same binary union" end text end define linebreak define pyk of lemma intersection subset as text "lemma intersection subset" end text end define linebreak define pyk of lemma same intersection as text "lemma same intersection" end text end define linebreak define pyk of lemma auto member as text "lemma auto member" end text end define linebreak define pyk of lemma eq-system not empty0 as text "lemma eq-system not empty0" end text end define linebreak define pyk of lemma eq-system not empty as text "lemma eq-system not empty" end text end define linebreak define pyk of lemma eq subset0 as text "lemma eq subset0" end text end define linebreak define pyk of lemma eq subset as text "lemma eq subset" end text end define linebreak define pyk of lemma equivalence nec condition0 as text "lemma equivalence nec condition0" end text end define linebreak define pyk of lemma equivalence nec condition as text "lemma equivalence nec condition" end text end define linebreak define pyk of lemma none-equivalence nec condition0 as text "lemma none-equivalence nec condition0" end text end define linebreak define pyk of lemma none-equivalence nec condition1 as text "lemma none-equivalence nec condition1" end text end define linebreak define pyk of lemma none-equivalence nec condition as text "lemma none-equivalence nec condition" end text end define linebreak define pyk of lemma equivalence class is subset as text "lemma equivalence class is subset" end text end define linebreak define pyk of lemma equivalence classes are disjoint as text "lemma equivalence classes are disjoint" end text end define linebreak define pyk of lemma all disjoint as text "lemma all disjoint" end text end define linebreak define pyk of lemma all disjoint-imply as text "lemma all disjoint-imply" end text end define linebreak define pyk of lemma bs subset union(bs/r) as text "lemma bs subset union(bs/r)" end text end define linebreak define pyk of lemma union(bs/r) subset bs as text "lemma union(bs/r) subset bs" end text end define linebreak define pyk of lemma union(bs/r) is bs as text "lemma union(bs/r) is bs" end text end define linebreak define pyk of theorem eq-system is partition as text "theorem eq-system is partition" end text end define linebreak define pyk of eq-system of x modulo x as text "eq-system of "! modulo "!" end text end define linebreak define pyk of intersection x comma x end intersection as text "intersection "! comma "! end intersection" end text end define linebreak define pyk of union x end union as text "union "! end union" end text end define linebreak define pyk of binary-union x comma x end union as text "binary-union "! comma "! end union" end text end define linebreak define pyk of power x end power as text "power "! end power" end text end define linebreak define pyk of zermelo singleton x end singleton as text "zermelo singleton "! end singleton" end text end define linebreak define pyk of zermelo pair x comma x end pair as text "zermelo pair "! comma "! end pair" end text end define linebreak define pyk of zermelo ordered pair x comma x end pair as text "zermelo ordered pair "! comma "! end pair" end text end define linebreak define pyk of x zermelo in x as text ""! zermelo in "!" end text end define linebreak define pyk of x is related to x under x as text ""! is related to "! under "!" end text end define linebreak define pyk of x is reflexive relation in x as text ""! is reflexive relation in "!" end text end define linebreak define pyk of x is symmetric relation in x as text ""! is symmetric relation in "!" end text end define linebreak define pyk of x is transitive relation in x as text ""! is transitive relation in "!" end text end define linebreak define pyk of x is equivalence relation in x as text ""! is equivalence relation in "!" end text end define linebreak define pyk of equivalence class of x in x modulo x as text "equivalence class of "! in "! modulo "!" end text end define linebreak define pyk of x is partition of x as text ""! is partition of "!" end text end define linebreak define pyk of x zermelo is x as text ""! zermelo is "!" end text end define linebreak define pyk of x is subset of x as text ""! is subset of "!" end text end define linebreak define pyk of not0 x as text "not0 "!" end text end define linebreak define pyk of x zermelo ~in x as text ""! zermelo ~in "!" end text end define linebreak define pyk of x zermelo ~is x as text ""! zermelo ~is "!" end text end define linebreak define pyk of x and0 x as text ""! and0 "!" end text end define linebreak define pyk of x or0 x as text ""! or0 "!" end text end define linebreak define pyk of x iff x as text ""! iff "!" end text end define linebreak define pyk of the set of ph in x such that x end set as text "the set of ph in "! such that "! end set" end text end define linebreak define pyk of equivalence-relations as text "equivalence-relations" end text end define linebreak unicode end of text end math ] "
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\section{Prioritetstabel} \label{sec:prio}
Den nedenst\aa{}ende tabel indeholder alle de formelle konstruktioner, som er til r\aa{}dighed i dette dokument. Konstruktionerne er grupperet efter prioritet; de f\o{}rstn\ae{}vnte grupper har st\o{}rst prioritet. Hver gruppe er markeret med et af de to ord ``$\mathbf{Preassociative}$'' eller ``$\mathbf{Postassociative}$'', der angiver, om konstruktionerne er venstre- eller h\o{}jreassociative.
" [ flush left math define priority of equivalence-relations as preassociative priority equivalence-relations equal priority base equal priority bracket x end bracket equal priority big bracket x end bracket equal priority math x end math equal priority flush left x end left equal priority var x equal priority var y equal priority var z equal priority proclaim x as x end proclaim equal priority define x of x as x end define equal priority pyk equal priority tex equal priority tex name equal priority priority equal priority x equal priority true equal priority if x then x else x end if equal priority introduce x of x as x end introduce equal priority value equal priority claim equal priority bottom equal priority function f of x end function equal priority identity x end identity equal priority false equal priority untagged zero equal priority untagged one equal priority untagged two equal priority untagged three equal priority untagged four equal priority untagged five equal priority untagged six equal priority untagged seven equal priority untagged eight equal priority untagged nine equal priority zero equal priority one equal priority two equal priority three equal priority four equal priority five equal priority six equal priority seven equal priority eight equal priority nine equal priority var a equal priority var b equal priority var c equal priority var d equal priority var e equal priority var f equal priority var g equal priority var h equal priority var i equal priority var j equal priority var k equal priority var l equal priority var m equal priority var n equal priority var o equal priority var p equal priority var q equal priority var r equal priority var s equal priority var t equal priority var u equal priority var v equal priority var w equal priority tagged parenthesis x end tagged equal priority tagged if x then x else x end if equal priority array x is x end array equal priority left equal priority center equal priority right equal priority empty equal priority substitute x set x to x end substitute equal priority map tag x end tag equal priority raw map untag x end untag equal priority map untag x end untag equal priority normalizing untag x end untag equal priority apply x to x end apply equal priority apply one x to x end apply equal priority identifier x end identifier equal priority identifier one x plus id x end identifier equal priority array plus x and x end plus equal priority array remove x array x level x end remove equal priority array put x value x array x level x end put equal priority array add x value x index x value x level x end add equal priority bit x of x end bit equal priority bit one x of x end bit equal priority example rack equal priority vector hook equal priority bibliography hook equal priority dictionary hook equal priority body hook equal priority codex hook equal priority expansion hook equal priority code hook equal priority cache hook equal priority diagnose hook equal priority pyk aspect equal priority tex aspect equal priority texname aspect equal priority value aspect equal priority message aspect equal priority macro aspect equal priority definition aspect equal priority unpack aspect equal priority claim aspect equal priority priority aspect equal priority lambda identifier equal priority apply identifier equal priority true identifier equal priority if identifier equal priority quote identifier equal priority proclaim identifier equal priority define identifier equal priority introduce identifier equal priority hide identifier equal priority pre identifier equal priority post identifier equal priority eval x stack x cache x end eval equal priority eval two x ref x id x stack x cache x end eval equal priority eval three x function x stack x cache x end eval equal priority eval four x arguments x stack x cache x end eval equal priority lookup x stack x default x end lookup equal priority abstract x term x stack x cache x end abstract equal priority quote x end quote equal priority expand x state x cache x end expand equal priority expand two x definition x state x cache x end expand equal priority expand list x state x cache x end expand equal priority macro equal priority macro state equal priority zip x with x end zip equal priority assoc one x address x index x end assoc equal priority protect x end protect equal priority self equal priority macro define x as x end define equal priority value define x as x end define equal priority intro define x as x end define equal priority pyk define x as x end define equal priority tex define x as x end define equal priority tex name define x as x end define equal priority priority table x end table equal priority macro define one equal priority macro define two x end define equal priority macro define three x end define equal priority macro define four x state x cache x definition x end define equal priority state expand x state x cache x end expand equal priority quote expand x term x stack x end expand equal priority quote expand two x term x stack x end expand equal priority quote expand three x term x stack x value x end expand equal priority quote expand star x term x stack x end expand equal priority parenthesis x end parenthesis equal priority big parenthesis x end parenthesis equal priority display x end display equal priority statement x end statement equal priority spying test x end test equal priority false spying test x end test equal priority aspect x subcodex x end aspect equal priority aspect x term x cache x end aspect equal priority tuple x end tuple equal priority tuple one x end tuple equal priority tuple two x end tuple equal priority let two x apply x end let equal priority let one x apply x end let equal priority claim define x as x end define equal priority checker equal priority check x cache x end check equal priority check two x cache x def x end check equal priority check three x cache x def x end check equal priority check list x cache x end check equal priority check list two x cache x value x end check equal priority test x end test equal priority false test x end test equal priority raw test x end test equal priority message equal priority message define x as x end define equal priority the statement aspect equal priority statement equal priority statement define x as x end define equal priority example axiom equal priority example scheme equal priority example rule equal priority absurdity equal priority contraexample equal priority example theory primed equal priority example lemma equal priority metavar x end metavar equal priority meta a equal priority meta b equal priority meta c equal priority meta d equal priority meta e equal priority meta f equal priority meta g equal priority meta h equal priority meta i equal priority meta j equal priority meta k equal priority meta l equal priority meta m equal priority meta n equal priority meta o equal priority meta p equal priority meta q equal priority meta r equal priority meta s equal priority meta t equal priority meta u equal priority meta v equal priority meta w equal priority meta x equal priority meta y equal priority meta z equal priority sub x set x to x end sub equal priority sub star x set x to x end sub equal priority the empty set equal priority example remainder equal priority make visible x end visible equal priority intro x index x pyk x tex x end intro equal priority intro x pyk x tex x end intro equal priority error x term x end error equal priority error two x term x end error equal priority proof x term x cache x end proof equal priority proof two x term x end proof equal priority sequent eval x term x end eval equal priority seqeval init x term x end eval equal priority seqeval modus x term x end eval equal priority seqeval modus one x term x sequent x end eval equal priority seqeval verify x term x end eval equal priority seqeval verify one x term x sequent x end eval equal priority sequent eval plus x term x end eval equal priority seqeval plus one x term x sequent x end eval equal priority seqeval minus x term x end eval equal priority seqeval minus one x term x sequent x end eval equal priority seqeval deref x term x end eval equal priority seqeval deref one x term x sequent x end eval equal priority seqeval deref two x term x sequent x def x end eval equal priority seqeval at x term x end eval equal priority seqeval at one x term x sequent x end eval equal priority seqeval infer x term x end eval equal priority seqeval infer one x term x premise x sequent x end eval equal priority seqeval endorse x term x end eval equal priority seqeval endorse one x term x side x sequent x end eval equal priority seqeval est x term x end eval equal priority seqeval est one x term x name x sequent x end eval equal priority seqeval est two x term x name x sequent x def x end eval equal priority seqeval all x term x end eval equal priority seqeval all one x term x variable x sequent x end eval equal priority seqeval cut x term x end eval equal priority seqeval cut one x term x forerunner x end eval equal priority seqeval cut two x term x forerunner x sequent x end eval equal priority computably true x end true equal priority claims x cache x ref x end claims equal priority claims two x cache x ref x end claims equal priority the proof aspect equal priority proof equal priority lemma x says x end lemma equal priority proof of x reads x end proof equal priority in theory x lemma x says x end lemma equal priority in theory x antilemma x says x end antilemma equal priority in theory x rule x says x end rule equal priority in theory x antirule x says x end antirule equal priority verifier equal priority verify one x end verify equal priority verify two x proofs x end verify equal priority verify three x ref x sequents x diagnose x end verify equal priority verify four x premises x end verify equal priority verify five x ref x array x sequents x end verify equal priority verify six x ref x list x sequents x end verify equal priority verify seven x ref x id x sequents x end verify equal priority cut x and x end cut equal priority head x end head equal priority tail x end tail equal priority rule one x theory x end rule equal priority rule x subcodex x end rule equal priority rule tactic equal priority plus x and x end plus equal priority theory x end theory equal priority theory two x cache x end theory equal priority theory three x name x end theory equal priority theory four x name x sum x end theory equal priority example axiom lemma primed equal priority example scheme lemma primed equal priority example rule lemma primed equal priority contraexample lemma primed equal priority example axiom lemma equal priority example scheme lemma equal priority example rule lemma equal priority contraexample lemma equal priority example theory equal priority ragged right equal priority ragged right expansion equal priority parameter term x stack x seed x end parameter equal priority parameter term star x stack x seed x end parameter equal priority instantiate x with x end instantiate equal priority instantiate star x with x end instantiate equal priority occur x in x substitution x end occur equal priority occur star x in x substitution x end occur equal priority unify x with x substitution x end unify equal priority unify star x with x substitution x end unify equal priority unify two x with x substitution x end unify equal priority ell a equal priority ell b equal priority ell c equal priority ell d equal priority ell e equal priority ell f equal priority ell g equal priority ell h equal priority ell i equal priority ell j equal priority ell k equal priority ell l equal priority ell m equal priority ell n equal priority ell o equal priority ell p equal priority ell q equal priority ell r equal priority ell s equal priority ell t equal priority ell u equal priority ell v equal priority ell w equal priority ell x equal priority ell y equal priority ell z equal priority ell big a equal priority ell big b equal priority ell big c equal priority ell big d equal priority ell big e equal priority ell big f equal priority ell big g equal priority ell big h equal priority ell big i equal priority ell big j equal priority ell big k equal priority ell big l equal priority ell big m equal priority ell big n equal priority ell big o equal priority ell big p equal priority ell big q equal priority ell big r equal priority ell big s equal priority ell big t equal priority ell big u equal priority ell big v equal priority ell big w equal priority ell big x equal priority ell big y equal priority ell big z equal priority ell dummy equal priority sequent reflexivity equal priority tactic reflexivity equal priority sequent commutativity equal priority tactic commutativity equal priority the tactic aspect equal priority tactic equal priority tactic define x as x end define equal priority proof expand x state x cache x end expand equal priority proof expand list x state x cache x end expand equal priority proof state equal priority conclude one x cache x end conclude equal priority conclude two x proves x cache x end conclude equal priority conclude three x proves x lemma x substitution x end conclude equal priority conclude four x lemma x end conclude equal priority check equal priority general macro define x as x end define equal priority make root visible x end visible equal priority sequent example axiom equal priority sequent example rule equal priority sequent example contradiction equal priority sequent example theory equal priority sequent example lemma equal priority set x end set equal priority object var x end var equal priority object a equal priority object b equal priority object c equal priority object d equal priority object e equal priority object f equal priority object g equal priority object h equal priority object i equal priority object j equal priority object k equal priority object l equal priority object m equal priority object n equal priority object o equal priority object p equal priority object q equal priority object r equal priority object s equal priority object t equal priority object u equal priority object v equal priority object w equal priority object x equal priority object y equal priority object z equal priority sub x is x where x is x end sub equal priority sub zero x is x where x is x end sub equal priority sub one x is x where x is x end sub equal priority sub star x is x where x is x end sub equal priority deduction x conclude x end deduction equal priority deduction zero x conclude x end deduction equal priority deduction one x conclude x condition x end deduction equal priority deduction two x conclude x condition x end deduction equal priority deduction three x conclude x condition x bound x end deduction equal priority deduction four x conclude x condition x bound x end deduction equal priority deduction four star x conclude x condition x bound x end deduction equal priority deduction five x condition x bound x end deduction equal priority deduction six x conclude x exception x bound x end deduction equal priority deduction six star x conclude x exception x bound x end deduction equal priority deduction seven x end deduction equal priority deduction eight x bound x end deduction equal priority deduction eight star x bound x end deduction equal priority system s equal priority double negation equal priority rule mp equal priority rule gen equal priority deduction equal priority axiom s one equal priority axiom s two equal priority axiom s three equal priority axiom s four equal priority axiom s five equal priority axiom s six equal priority axiom s seven equal priority axiom s eight equal priority axiom s nine equal priority repetition equal priority lemma a one equal priority lemma a two equal priority lemma a four equal priority lemma a five equal priority prop three two a equal priority prop three two b equal priority prop three two c equal priority prop three two d equal priority prop three two e one equal priority prop three two e two equal priority prop three two e equal priority prop three two f one equal priority prop three two f two equal priority prop three two f equal priority prop three two g one equal priority prop three two g two equal priority prop three two g equal priority prop three two h one equal priority prop three two h two equal priority prop three two h equal priority block one x state x cache x end block equal priority block two x end block equal priority cdots equal priority object-var equal priority ex-var equal priority ph-var equal priority vaerdi equal priority variabel equal priority op x end op equal priority op2 x comma x end op2 equal priority define-equal x comma x end equal equal priority contains-empty x end empty equal priority 1deduction x conclude x end 1deduction equal priority 1deduction zero x conclude x end 1deduction equal priority 1deduction side x conclude x condition x end 1deduction equal priority 1deduction one x conclude x condition x end 1deduction equal priority 1deduction two x conclude x condition x end 1deduction equal priority 1deduction three x conclude x condition x bound x end 1deduction equal priority 1deduction four x conclude x condition x bound x end 1deduction equal priority 1deduction four star x conclude x condition x bound x end 1deduction equal priority 1deduction five x condition x bound x end 1deduction equal priority 1deduction six x conclude x exception x bound x end 1deduction equal priority 1deduction six star x conclude x exception x bound x end 1deduction equal priority 1deduction seven x end 1deduction equal priority 1deduction eight x bound x end 1deduction equal priority 1deduction eight star x bound x end 1deduction equal priority ex1 equal priority ex2 equal priority ex10 equal priority ex20 equal priority existential var x end var equal priority x is existential var equal priority exist-sub x is x where x is x end sub equal priority exist-sub0 x is x where x is x end sub equal priority exist-sub1 x is x where x is x end sub equal priority exist-sub* x is x where x is x end sub equal priority placeholder-var1 equal priority placeholder-var2 equal priority placeholder-var3 equal priority placeholder-var x end var equal priority x is placeholder-var equal priority ph-sub x is x where x is x end sub equal priority ph-sub0 x is x where x is x end sub equal priority ph-sub1 x is x where x is x end sub equal priority ph-sub* x is x where x is x end sub equal priority var big set equal priority object big set equal priority meta big set equal priority zermelo empty set equal priority system zf equal priority 1rule mp equal priority 1rule gen equal priority 1rule repetition equal priority 1rule ad absurdum equal priority 1rule deduction equal priority 1rule exist intro equal priority axiom extensionality equal priority axiom empty set equal priority axiom pair definition equal priority axiom union definition equal priority axiom power definition equal priority axiom separation definition equal priority cheating axiom all disjoint equal priority prop lemma add double neg equal priority prop lemma remove double neg equal priority prop lemma and commutativity equal priority prop lemma auto imply equal priority prop lemma contrapositive equal priority prop lemma first conjunct equal priority prop lemma second conjunct equal priority prop lemma from contradiction equal priority prop lemma from disjuncts equal priority prop lemma iff commutativity equal priority prop lemma iff first equal priority prop lemma iff second equal priority prop lemma imply transitivity equal priority prop lemma join conjuncts equal priority prop lemma mp2 equal priority prop lemma mp3 equal priority prop lemma mp4 equal priority prop lemma mp5 equal priority prop lemma mt equal priority prop lemma negative mt equal priority prop lemma technicality equal priority prop lemma weakening equal priority prop lemma weaken or first equal priority prop lemma weaken or second equal priority lemma formula2pair equal priority lemma pair2formula equal priority lemma formula2union equal priority lemma union2formula equal priority lemma formula2separation equal priority lemma separation2formula equal priority lemma subset in power set equal priority lemma power set is subset0 equal priority lemma power set is subset equal priority lemma power set is subset0-switch equal priority lemma power set is subset-switch equal priority lemma set equality suff condition equal priority lemma set equality suff condition(t)0 equal priority lemma set equality suff condition(t) equal priority lemma set equality skip quantifier equal priority lemma set equality nec condition equal priority lemma reflexivity0 equal priority lemma reflexivity equal priority lemma symmetry0 equal priority lemma symmetry equal priority lemma transitivity0 equal priority lemma transitivity equal priority lemma er is reflexive equal priority lemma er is symmetric equal priority lemma er is transitive equal priority lemma empty set is subset equal priority lemma member not empty0 equal priority lemma member not empty equal priority lemma unique empty set0 equal priority lemma unique empty set equal priority lemma =reflexivity equal priority lemma =symmetry equal priority lemma =transitivity0 equal priority lemma =transitivity equal priority lemma transfer ~is0 equal priority lemma transfer ~is equal priority lemma pair subset0 equal priority lemma pair subset1 equal priority lemma pair subset equal priority lemma same pair equal priority lemma same singleton equal priority lemma union subset equal priority lemma same union equal priority lemma separation subset equal priority lemma same separation equal priority lemma same binary union equal priority lemma intersection subset equal priority lemma same intersection equal priority lemma auto member equal priority lemma eq-system not empty0 equal priority lemma eq-system not empty equal priority lemma eq subset0 equal priority lemma eq subset equal priority lemma equivalence nec condition0 equal priority lemma equivalence nec condition equal priority lemma none-equivalence nec condition0 equal priority lemma none-equivalence nec condition1 equal priority lemma none-equivalence nec condition equal priority lemma equivalence class is subset equal priority lemma equivalence classes are disjoint equal priority lemma all disjoint equal priority lemma all disjoint-imply equal priority lemma bs subset union(bs/r) equal priority lemma union(bs/r) subset bs equal priority lemma union(bs/r) is bs equal priority theorem eq-system is partition end priority greater than preassociative priority x sub x end sub equal priority x intro x index x pyk x tex x end intro equal priority x intro x pyk x tex x end intro equal priority x intro x index x pyk x tex x name x end intro equal priority x intro x pyk x tex x name x end intro equal priority x prime equal priority x assoc x end assoc equal priority x set x to x end set equal priority x set multi x to x end set equal priority x bit nil equal priority x bit one equal priority binary equal priority x color x end color equal priority x color star x end color equal priority x raw head equal priority x raw tail equal priority x cardinal untag equal priority x head equal priority x tail equal priority x is singular equal priority x is cardinal equal priority x is data equal priority x is atomic equal priority x cardinal retract equal priority x tagged retract equal priority x boolean retract equal priority x ref equal priority x id equal priority x debug equal priority x root equal priority x zeroth equal priority x first equal priority x second equal priority x third equal priority x fourth equal priority x fifth equal priority x sixth equal priority x seventh equal priority x eighth equal priority x ninth equal priority x is error equal priority x is metavar equal priority x is metaclosed equal priority x is metaclosed star equal priority x hide end priority greater than preassociative priority unicode start of text x end unicode text equal priority unicode end of text equal priority text x end text equal priority text x plus x equal priority text x plus indent x equal priority unicode newline x equal priority unicode space x equal priority unicode exclamation mark x equal priority unicode quotation mark x equal priority unicode number sign x equal priority unicode dollar sign x equal priority unicode percent x equal priority unicode ampersand x equal priority unicode apostrophe x equal priority unicode left parenthesis x equal priority unicode right parenthesis x equal priority unicode asterisk x equal priority unicode plus sign x equal priority unicode comma x equal priority unicode hyphen x equal priority unicode period x equal priority unicode slash x equal priority unicode zero x equal priority unicode one x equal priority unicode two x equal priority unicode three x equal priority unicode four x equal priority unicode five x equal priority unicode six x equal priority unicode seven x equal priority unicode eight x equal priority unicode nine x equal priority unicode colon x equal priority unicode semicolon x equal priority unicode less than x equal priority unicode equal sign x equal priority unicode greater than x equal priority unicode question mark x equal priority unicode commercial at x equal priority unicode capital a x equal priority unicode capital b x equal priority unicode capital c x equal priority unicode capital d x equal priority unicode capital e x equal priority unicode capital f x equal priority unicode capital g x equal priority unicode capital h x equal priority unicode capital i x equal priority unicode capital j x equal priority unicode capital k x equal priority unicode capital l x equal priority unicode capital m x equal priority unicode capital n x equal priority unicode capital o x equal priority unicode capital p x equal priority unicode capital q x equal priority unicode capital r x equal priority unicode capital s x equal priority unicode capital t x equal priority unicode capital u x equal priority unicode capital v x equal priority unicode capital w x equal priority unicode capital x x equal priority unicode capital y x equal priority unicode capital z x equal priority unicode left bracket x equal priority unicode backslash x equal priority unicode right bracket x equal priority unicode circumflex x equal priority unicode underscore x equal priority unicode grave accent x equal priority unicode small a x equal priority unicode small b x equal priority unicode small c x equal priority unicode small d x equal priority unicode small e x equal priority unicode small f x equal priority unicode small g x equal priority unicode small h x equal priority unicode small i x equal priority unicode small j x equal priority unicode small k x equal priority unicode small l x equal priority unicode small m x equal priority unicode small n x equal priority unicode small o x equal priority unicode small p x equal priority unicode small q x equal priority unicode small r x equal priority unicode small s x equal priority unicode small t x equal priority unicode small u x equal priority unicode small v x equal priority unicode small w x equal priority unicode small x x equal priority unicode small y x equal priority unicode small z x equal priority unicode left brace x equal priority unicode vertical line x equal priority unicode right brace x equal priority unicode tilde x equal priority preassociative x greater than x equal priority postassociative x greater than x equal priority priority x equal x equal priority priority x end priority equal priority newline x equal priority macro newline x equal priority macro indent x end priority greater than preassociative priority x apply x equal priority x tagged apply x end priority greater than preassociative priority x suc end priority greater than preassociative priority eq-system of x modulo x equal priority intersection x comma x end intersection end priority greater than preassociative priority union x end union equal priority binary-union x comma x end union equal priority power x end power end priority greater than preassociative priority zermelo singleton x end singleton end priority greater than preassociative priority zermelo pair x comma x end pair equal priority zermelo ordered pair x comma x end pair end priority greater than preassociative priority x zermelo in x equal priority x is related to x under x equal priority x is reflexive relation in x equal priority x is symmetric relation in x equal priority x is transitive relation in x equal priority x is equivalence relation in x equal priority equivalence class of x in x modulo x equal priority x is partition of x end priority greater than preassociative priority x times x equal priority x times zero x end priority greater than preassociative priority x plus x equal priority x plus zero x equal priority x plus one x equal priority x minus x equal priority x minus zero x equal priority x minus one x end priority greater than preassociative priority x term plus x end plus equal priority x term union x equal priority x term minus x end minus end priority greater than postassociative priority x raw pair x equal priority x eager pair x equal priority x tagged pair x equal priority x untagged double x equal priority x pair x equal priority x double x end priority greater than postassociative priority x comma x end priority greater than preassociative priority x boolean equal x equal priority x data equal x equal priority x cardinal equal x equal priority x peano equal x equal priority x tagged equal x equal priority x math equal x equal priority x reduce to x equal priority x term equal x equal priority x term list equal x equal priority x term root equal x equal priority x term in x equal priority x term subset x equal priority x term set equal x equal priority x sequent equal x equal priority x free in x equal priority x free in star x equal priority x free for x in x equal priority x free for star x in x equal priority x claim in x equal priority x less x equal priority x less zero x equal priority x less one x equal priority x equal x equal priority x unequal x equal priority x is object var equal priority x avoid zero x equal priority x avoid one x equal priority x avoid star x equal priority x zermelo is x equal priority x is subset of x end priority greater than preassociative priority not x equal priority not0 x equal priority x zermelo ~in x equal priority x zermelo ~is x end priority greater than preassociative priority x and x equal priority x macro and x equal priority x simple and x equal priority x claim and x equal priority x and0 x end priority greater than preassociative priority x or x equal priority x parallel x equal priority x macro or x equal priority x or0 x end priority greater than preassociative priority exist x indeed x equal priority for all x indeed x equal priority for all objects x indeed x end priority greater than postassociative priority x macro imply x equal priority x imply x equal priority x if and only if x equal priority x iff x end priority greater than preassociative priority the set of ph in x such that x end set end priority greater than postassociative priority x guard x equal priority x spy x equal priority x tagged guard x end priority greater than preassociative priority x select x else x end select end priority greater than preassociative priority lambda x dot x equal priority tagged lambda x dot x equal priority tagging x equal priority open if x then x else x equal priority let x be x in x equal priority let x abbreviate x in x end priority greater than preassociative priority x avoid x end priority greater than preassociative priority x init equal priority x modus equal priority x verify equal priority x curry plus equal priority x curry minus equal priority x dereference end priority greater than preassociative priority x at x equal priority x modus ponens x equal priority x modus probans x equal priority x conclude x equal priority x object modus ponens x end priority greater than postassociative priority x infer x equal priority x endorse x equal priority x id est x end priority greater than preassociative priority all x indeed x equal priority for all terms x indeed x end priority greater than postassociative priority x rule plus x end priority greater than postassociative priority x cut x end priority greater than preassociative priority x proves x end priority greater than preassociative priority x proof of x reads x equal priority line x because x indeed x end line x equal priority because x indeed x qed equal priority line x premise x end line x equal priority line x side condition x end line x equal priority arbitrary x end line x equal priority locally define x as x end line x equal priority block x line x end block x equal priority because x indeed x end line equal priority any term x end line x end priority greater than postassociative priority x alternative x end priority greater than postassociative priority x , x equal priority x [ x ] x end priority greater than preassociative priority x tab x end priority greater than preassociative priority x row x equal priority x linebreak x equal priority x safe row x end priority greater than unicode end of text end define end math end left ] "
\section{\TeX{} definitioner} \label{sec:tex}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define tex of equivalence-relations as text "EquivalenceRelations" end text end define end math ] "
\item " [ math define tex of cdots as text "(\cdots{})" end text end define end math ] "
\item " [ math define tex of object-var as text "\texttt{Objekt-var}" end text end define end math ] "
\item " [ math define tex of ex-var as text "\texttt{Ex-var}" end text end define end math ] "
\item " [ math define tex of ph-var as text "\texttt{Ph-var}" end text end define end math ] "
\item " [ math define tex of vaerdi as text "\texttt{V\ae{}rdi}" end text end define end math ] "
\item " [ math define tex of variabel as text "\texttt{Variabel}" end text end define end math ] "
\item " [ math define tex of op var x end op as text "Op(#1.
)" end text end define end math ] "
\item " [ math define tex of op2 var x comma var y end op2 as text "Op(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of define-equal var x comma var y end equal as text "#1.
\mathrel {\ddot {=}} #2." end text end define end math ] "
\item " [ math define tex of contains-empty var x end empty as text "ContainsEmpty(#1.
)" end text end define end math ] "
" [ math define tex of 1deduction var x conclude var y end 1deduction as text "
Dedu(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction zero var x conclude var y end 1deduction as text "
Dedu_0(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction side var x conclude var y condition var z end 1deduction as text "Dedu_{s}(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction one var x conclude var y condition var z end 1deduction as text "
Dedu_1(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction two var x conclude var y condition var z end 1deduction as text "
Dedu_2(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction three var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_3(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction four var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_4(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction four star var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_4^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction five var x condition var y bound var z end 1deduction as text "
Dedu_5(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction six var p conclude var c exception var e bound var b end 1deduction as text "
Dedu_6(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction six star var p conclude var c exception var e bound var b end 1deduction as text "
Dedu_6^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction seven var p end 1deduction as text "
Dedu_7(#1.
)" end text end define end math ] "
" [ math define tex of 1deduction eight var p bound var b end 1deduction as text "
Dedu_8(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction eight star var p bound var b end 1deduction as text "
Dedu_8^*(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of ex1 as text "Ex_{1}" end text end define end math ] "
\item " [ math define tex of ex2 as text "Ex_{2}" end text end define end math ] "
\item " [ math define tex of ex10 as text "Ex_{10}" end text end define end math ] "
\item " [ math define tex of ex20 as text "Ex_{20}" end text end define end math ] "
\item " [ math define tex of existential var var x end var as text "#1.
_{Ex}" end text end define end math ] "
\item " [ math define tex of var x is existential var as text "#1.
^{Ex}" end text end define end math ] "
" [ math define tex of exist-sub var x is var y where var z is var u end sub as text "\langle #1.
{\equiv} #2.
| #3.
{:=} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub0 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^0 #2.
| #3.
{:=} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub1 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^1 #2.
| #3.
{:=} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub* var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^* #2.
| #3.
{:=} #4.
\rangle_{Ex} " end text end define end math ] "
\item " [ math define tex of placeholder-var1 as text "ph_{1}" end text end define end math ] "
\item " [ math define tex of placeholder-var2 as text "ph_{2}" end text end define end math ] "
\item " [ math define tex of placeholder-var3 as text "ph_{3}" end text end define end math ] "
\item " [ math define tex of placeholder-var var x end var as text "#1.
_{Ph} " end text end define end math ] "
\item " [ math define tex of var x is placeholder-var as text "#1.
^{Ph}" end text end define end math ] "
" [ math define tex of ph-sub var x is var y where var z is var u end sub as text "\langle #1.
{\equiv} #2.
| #3.
{:=} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub0 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^0 #2.
| #3.
{:=} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub1 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^1 #2.
| #3.
{:=} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub* var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^* #2.
| #3.
{:=} #4.
\rangle_{Ph} " end text end define end math ] "
\item " [ math define tex of var big set as text "\mathsf {bs}" end text end define end math ] "
\item " [ math define tex of object big set as text " \mathsf {OBS}" end text end define end math ] "
" [ math define tex of meta big set as text "{\cal BS}" end text end define end math ] "
\item " [ math define tex of zermelo empty set as text "\mathrm{\O}" end text end define end math ] "
\item " [ math define tex of system zf as text "ZFsub" end text end define end math ] "
\item " [ math define tex of 1rule mp as text "MP" end text end define end math ] "
\item " [ math define tex of 1rule gen as text "Gen" end text end define end math ] "
\item " [ math define tex of 1rule repetition as text "Repetition" end text end define end math ] "
\item " [ math define tex of 1rule ad absurdum as text "Neg" end text end define end math ] "
\item " [ math define tex of 1rule deduction as text "Ded" end text end define end math ] "
\item " [ math define tex of 1rule exist intro as text "ExistIntro" end text end define end math ] "
\item " [ math define tex of axiom extensionality as text "Extensionality" end text end define end math ] "
\item " [ math define tex of axiom empty set as text "\O{}def" end text end define end math ] "
\item " [ math define tex of axiom pair definition as text "PairDef" end text end define end math ] "
\item " [ math define tex of axiom union definition as text "UnionDef" end text end define end math ] "
\item " [ math define tex of axiom power definition as text "PowerDef" end text end define end math ] "
\item " [ math define tex of axiom separation definition as text "SeparationDef" end text end define end math ] "
\item " [ math define tex of cheating axiom all disjoint as text "CheatAllDisjoint" end text end define end math ] "
\item " [ math define tex of prop lemma add double neg as text "AddDoubleNeg" end text end define end math ] "
\item " [ math define tex of prop lemma remove double neg as text "RemoveDoubleNeg" end text end define end math ] "
\item " [ math define tex of prop lemma and commutativity as text "AndCommutativity" end text end define end math ] "
\item " [ math define tex of prop lemma auto imply as text "AutoImply" end text end define end math ] "
\item " [ math define tex of prop lemma contrapositive as text "Contrapositive" end text end define end math ] "
\item " [ math define tex of prop lemma first conjunct as text "FirstConjunct" end text end define end math ] "
\item " [ math define tex of prop lemma second conjunct as text "SecondConjunct" end text end define end math ] "
\item " [ math define tex of prop lemma from contradiction as text "FromContradiction" end text end define end math ] "
\item " [ math define tex of prop lemma from disjuncts as text "FromDisjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma iff commutativity as text "IffCommutativity" end text end define end math ] "
\item " [ math define tex of prop lemma iff first as text "IffFirst" end text end define end math ] "
\item " [ math define tex of prop lemma iff second as text "IffSecond" end text end define end math ] "
\item " [ math define tex of prop lemma imply transitivity as text "ImplyTransitivity" end text end define end math ] "
\item " [ math define tex of prop lemma join conjuncts as text "JoinConjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma mp2 as text "MP2" end text end define end math ] "
\item " [ math define tex of prop lemma mp3 as text "MP3" end text end define end math ] "
\item " [ math define tex of prop lemma mp4 as text "MP4" end text end define end math ] "
\item " [ math define tex of prop lemma mp5 as text "MP5" end text end define end math ] "
\item " [ math define tex of prop lemma mt as text "MT" end text end define end math ] "
\item " [ math define tex of prop lemma negative mt as text "NegativeMT" end text end define end math ] "
\item " [ math define tex of prop lemma technicality as text "Technicality" end text end define end math ] "
\item " [ math define tex of prop lemma weakening as text "Weakening" end text end define end math ] "
\item " [ math define tex of prop lemma weaken or first as text "WeakenOr1" end text end define end math ] "
\item " [ math define tex of prop lemma weaken or second as text "WeakenOr2" end text end define end math ] "
\item " [ math define tex of lemma pair2formula as text "Pair2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2pair as text "Formula2Pair" end text end define end math ] "
\item " [ math define tex of lemma union2formula as text "Union2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2union as text "Formula2Union" end text end define end math ] "
\item " [ math define tex of lemma separation2formula as text "Sep2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2separation as text "Formula2Sep" end text end define end math ] "
\item " [ math define tex of lemma subset in power set as text "SubsetInPower" end text end define end math ] "
\item " [ math define tex of lemma power set is subset0 as text "HelperPowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset as text "PowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset0-switch as text "(Switch)HelperPowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset-switch as text "(Switch)PowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition as text "ToSetEquality" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition(t)0 as text "HelperToSetEquality(t)" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition(t) as text "ToSetEquality(t)" end text end define end math ] "
\item " [ math define tex of lemma set equality skip quantifier as text "HelperFromSetEquality" end text end define end math ] "
\item " [ math define tex of lemma set equality nec condition as text "FromSetEquality" end text end define end math ] "
\item " [ math define tex of lemma reflexivity0 as text "HelperReflexivity" end text end define end math ] "
\item " [ math define tex of lemma reflexivity as text "Reflexivity" end text end define end math ] "
\item " [ math define tex of lemma symmetry0 as text "HelperSymmetry" end text end define end math ] "
\item " [ math define tex of lemma symmetry as text "Symmetry" end text end define end math ] "
\item " [ math define tex of lemma transitivity0 as text "HelperTransitivity" end text end define end math ] "
\item " [ math define tex of lemma transitivity as text "Transitivity" end text end define end math ] ",
\item " [ math define tex of lemma er is reflexive as text "ERisReflexive" end text end define end math ] "
\item " [ math define tex of lemma er is symmetric as text "ERisSymmetric" end text end define end math ] "
\item " [ math define tex of lemma er is transitive as text "ERisTransitive" end text end define end math ] "
\item " [ math define tex of lemma empty set is subset as text "\O{}isSubset" end text end define end math ] "
\item " [ math define tex of lemma member not empty0 as text "HelperMemberNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma member not empty as text "MemberNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma unique empty set0 as text "HelperUnique\O{}" end text end define end math ] "
\item " [ math define tex of lemma unique empty set as text "Unique\O{}" end text end define end math ] "
\item " [ math define tex of lemma =reflexivity as text "=\!{}Reflexivity" end text end define end math ] "
\item " [ math define tex of lemma =symmetry as text "=\!{}Symmetry" end text end define end math ] "
\item " [ math define tex of lemma =transitivity0 as text "Helper\!{}=\!{}Transitivity" end text end define end math ] "
\item " [ math define tex of lemma =transitivity as text "\!{}=\!{}Transitivity" end text end define end math ] "
\item " [ math define tex of lemma transfer ~is0 as text "HelperTransferNotEq" end text end define end math ] "
\item " [ math define tex of lemma transfer ~is as text "TransferNotEq" end text end define end math ] "
\item " [ math define tex of lemma pair subset0 as text "HelperPairSubset" end text end define end math ] "
\item " [ math define tex of lemma pair subset1 as text "Helper(2)PairSubset" end text end define end math ] "
\item " [ math define tex of lemma pair subset as text "PairSubset" end text end define end math ] "
\item " [ math define tex of lemma same pair as text "SamePair" end text end define end math ] "
\item " [ math define tex of lemma same singleton as text "SameSingleton" end text end define end math ] "
\item " [ math define tex of lemma union subset as text "UnionSubset" end text end define end math ] "
\item " [ math define tex of lemma same union as text "SameUnion" end text end define end math ] "
\item " [ math define tex of lemma separation subset as text "SeparationSubset" end text end define end math ] "
\item " [ math define tex of lemma same separation as text "SameSeparation" end text end define end math ] "
\item " [ math define tex of lemma same binary union as text "SameBinaryUnion" end text end define end math ] "
\item " [ math define tex of lemma intersection subset as text "IntersectionSubset" end text end define end math ] "
\item " [ math define tex of lemma same intersection as text "SameIntersection" end text end define end math ] "
\item " [ math define tex of lemma auto member as text "AutoMember" end text end define end math ] "
\item " [ math define tex of lemma eq-system not empty0 as text "HelperEqSysNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma eq-system not empty as text "EqSysNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma eq subset0 as text "HelperEqSubset" end text end define end math ] "
\item " [ math define tex of lemma eq subset as text "EqSubset" end text end define end math ] "
\item " [ math define tex of lemma equivalence nec condition as text "EqNecessary" end text end define end math ] "
\item " [ math define tex of lemma equivalence nec condition0 as text "HelperEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition0 as text "HelperNoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition1 as text "Helper(2)NoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition as text "NoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma equivalence class is subset as text "EqClassIsSubset" end text end define end math ] "
\item " [ math define tex of lemma equivalence classes are disjoint as text "EqClassesAreDisjoint" end text end define end math ] "
\item " [ math define tex of lemma all disjoint as text "AllDisjoint" end text end define end math ] "
\item " [ math define tex of lemma all disjoint-imply as text "AllDisjointImply" end text end define end math ] "
\item " [ math define tex of lemma bs subset union(bs/r) as text "BSsubset" end text end define end math ] "
\item " [ math define tex of lemma union(bs/r) subset bs as text "Union(BS/R)subset" end text end define end math ] "
\item " [ math define tex of lemma union(bs/r) is bs as text "UnionIdentity" end text end define end math ] "
\item " [ math define tex of theorem eq-system is partition as text "EqSysIsPartition" end text end define end math ] "
\item " [ math define tex of eq-system of var x modulo var y as text "#1.
/ #2." end text end define end math ] "
\item " [ math define tex of intersection var x comma var y end intersection as text "#1.
\cap #2." end text end define end math ] "
\item " [ math define tex of union var x end union as text "\cup #1." end text end define end math ] "
\item " [ math define tex of binary-union var x comma var y end union as text "#1.
\mathrel{\cup} #2." end text end define end math ] "
\item " [ math define tex of power var x end power as text "P(#1.
)" end text end define end math ] "
\item " [ math define tex of zermelo singleton var x end singleton as text "\{#1.
\}" end text end define end math ] "
\item " [ math define tex of zermelo pair var x comma var y end pair as text "\{#1.
,#2.
\}" end text end define end math ] "
\item " [ math define tex of zermelo ordered pair var x comma var y end pair as text "\langle #1.
,#2.
\rangle" end text end define end math ] ",
\item " [ math define tex of var x zermelo in var y as text "#1.
\mathrel{\in} #2." end text end define end math ] "
\item " [ math define tex of var x is related to var y under var z as text "#3.
(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is reflexive relation in var x as text "ReflRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is symmetric relation in var x as text "SymRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is transitive relation in var x as text "TransRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is equivalence relation in var x as text "EqRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of equivalence class of var x in var big set modulo var r as text "[#1.
\mathrel{\in} #2.
]_{#3.
}" end text end define end math ] "
\item " [ math define tex of var x is partition of var y as text "Partition(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var x zermelo is var y as text "#1.
\!\mathrel{=}\! #2." end text end define end math ] "
\item " [ math define tex of var x is subset of var y as text "#1.
\mathrel{\subseteq} #2." end text end define end math ] "
\item " [ math define tex of not0 var x as text "\dot{\neg}\, #1." end text end define end math ] "
\item " [ math define tex of var x zermelo ~in var y as text "#1.
\mathrel{\notin} #2." end text end define end math ] "
\item " [ math define tex of var x zermelo ~is var y as text "#1.
\mathrel{\neq} #2." end text end define end math ] "
\item " [ math define tex of var x and0 var y as text "#1.
\mathrel{\dot{\wedge}} #2." end text end define end math ] "
\item " [ math define tex of var x or0 var y as text "#1.
\mathrel{\dot{\vee}} #2." end text end define end math ] "
" [ math define tex of var x iff var y as text "#1.
\mathrel{\dot{\Leftrightarrow}} #2." end text end define end math ] "
\item " [ math define tex of the set of ph in var x such that var a end set as text " \{ ph \mathrel{\in} #1.
\mid #2.
\}" end text end define end math ] "
\end{list}
\end{document}
End of file
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The pyk compiler, version 0.grue.20060417+ by Klaus Grue,