Logiweb codex of EquivalenceRelations

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x, EquivalenceRelations, (⋯), Objekt-var, Ex-var, Ph-var, Værdi, Variabel, Op(∗), Op(∗, ∗), ∗ ¨= ∗, ContainsEmpty(∗), Dedu(∗, ∗), Dedu_{0}(∗, ∗), Dedu_{s}(∗, ∗, ∗), Dedu_{1}(∗, ∗, ∗), Dedu_{2}(∗, ∗, ∗), Dedu_{3}(∗, ∗, ∗, ∗), Dedu_{4}(∗, ∗, ∗, ∗), Dedu_{4}^{∗}(∗, ∗, ∗, ∗), Dedu_{5}(∗, ∗, ∗), Dedu_{6}(∗, ∗, ∗, ∗), Dedu_{6}^{∗}(∗, ∗, ∗, ∗), Dedu_{7}(∗), Dedu_{8}(∗, ∗), Dedu_{8}^{∗}(∗, ∗), Ex_{1}, Ex_{2}, Ex_{10}, Ex_{20}, ∗_{Ex}, ∗^{Ex}, ⟨∗≡ ∗ | ∗:=∗⟩_{Ex}, ⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ex}, ⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ex}, ⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ex}, ph_{1}, ph_{2}, ph_{3}, ∗_{Ph}, ∗^{Ph}, ⟨∗≡ ∗ | ∗:=∗⟩_{Ph}, ⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ph}, ⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ph}, ⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ph}, bs, OBS, BS, Ø, ZFsub, MP, Gen, Repetition, Neg, Ded, ExistIntro, Extensionality, Ødef, PairDef, UnionDef, PowerDef, SeparationDef, CheatAllDisjoint, AddDoubleNeg, RemoveDoubleNeg, AndCommutativity, AutoImply, Contrapositive, FirstConjunct, SecondConjunct, FromContradiction, FromDisjuncts, IffCommutativity, IffFirst, IffSecond, ImplyTransitivity, JoinConjuncts, MP2, MP3, MP4, MP5, MT, NegativeMT, Technicality, Weakening, WeakenOr1, WeakenOr2, Formula2Pair, Pair2Formula, Formula2Union, Union2Formula, Formula2Sep, Sep2Formula, SubsetInPower, HelperPowerIsSub, PowerIsSub, (Switch)HelperPowerIsSub, (Switch)PowerIsSub, ToSetEquality, HelperToSetEquality(t), ToSetEquality(t), HelperFromSetEquality, FromSetEquality, HelperReflexivity, Reflexivity, HelperSymmetry, Symmetry, HelperTransitivity, Transitivity, ERisReflexive, ERisSymmetric, ERisTransitive, ØisSubset, HelperMemberNotØ, MemberNotØ, HelperUniqueØ, UniqueØ, =Reflexivity, =Symmetry, Helper=Transitivity, =Transitivity, HelperTransferNotEq, TransferNotEq, HelperPairSubset, Helper(2)PairSubset, PairSubset, SamePair, SameSingleton, UnionSubset, SameUnion, SeparationSubset, SameSeparation, SameBinaryUnion, IntersectionSubset, SameIntersection, AutoMember, HelperEqSysNotØ, EqSysNotØ, HelperEqSubset, EqSubset, HelperEqNecessary, EqNecessary, HelperNoneEqNecessary, Helper(2)NoneEqNecessary, NoneEqNecessary, EqClassIsSubset, EqClassesAreDisjoint, AllDisjoint, AllDisjointImply, BSsubset, Union(BS/R)subset, UnionIdentity, EqSysIsPartition, ∗/∗, ∗ ∩ ∗, ∪∗, ∗ ∪ ∗, P(∗), {∗}, {∗, ∗}, ⟨∗, ∗⟩, ∗ ∈ ∗, ∗(∗, ∗), ReflRel(∗, ∗), SymRel(∗, ∗), TransRel(∗, ∗), EqRel(∗, ∗), [∗ ∈ ∗]_{∗}, Partition(∗, ∗), ∗=∗, ∗ ⊆ ∗, ˙¬ ∗, ∗ ∉ ∗,

∗ ≠ ∗, ∗ ˙∧ ∗, ∗ ˙∨ ∗, ∗ ˙⇔ ∗, {ph ∈ ∗ | ∗},

x

[x →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ¨= y]⌉)]

[x →^{val} y]

EquivalenceRelations

[EquivalenceRelations →^{prio} Preassociative [EquivalenceRelations], [base], [bracket ∗ end bracket], [big bracket ∗ end bracket], [ $ ∗ $ ], [flush left [∗]], [x], [y], [z], [[∗ ▷◁ ∗]], [[∗ →^{∗} ∗]], [pyk], [tex], [name], [prio], [∗], [T], [if(∗, ∗, ∗)], [[∗ ⇒^{∗} ∗]], [val], [claim], [⊥], [f(∗)], [(∗)^{I}], [F], [\underline{0}], [\underline{1}], [\underline{2}], [\underline{3}], [\underline{4}], [\underline{5}], [\underline{6}], [\underline{7}], [\underline{8}], [\underline{9}], [0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [a], [b], [c], [d], [e], [f], [g], [h], [i], [j], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [(∗)^{M}], [If(∗, ∗, ∗)], [array{∗} ∗ end array], [l], [c], [r], [empty], [⟨∗ | ∗:= ∗⟩], [M(∗)], [Ũ(∗)], [U(∗)], [U^{M}(∗)], [apply(∗, ∗)], [apply_{1}(∗, ∗)], [identifier(∗)], [identifier_{1}(∗, ∗)], [array-plus(∗, ∗)], [array-remove(∗, ∗, ∗)], [array-put(∗, ∗, ∗, ∗)], [array-add(∗, ∗, ∗, ∗, ∗)], [bit(∗, ∗)], [bit_{1}(∗, ∗)], [rack], ["vector"], ["bibliography"], ["dictionary"], ["body"], ["codex"], ["expansion"], ["code"], ["cache"], ["diagnose"], ["pyk"], ["tex"], ["texname"], ["value"], ["message"], ["macro"], ["definition"], ["unpack"], ["claim"], ["priority"], ["lambda"], ["apply"], ["true"], ["if"], ["quote"], ["proclaim"], ["define"], ["introduce"], ["hide"], ["pre"], ["post"], [E(∗, ∗, ∗)], [E_{2}(∗, ∗, ∗, ∗, ∗)], [E_{3}(∗, ∗, ∗, ∗)], [E_{4}(∗, ∗, ∗, ∗)], [lookup(∗, ∗, ∗)], [abstract(∗, ∗, ∗, ∗)], [⌈∗⌉], [M(∗, ∗, ∗)], [M_{2}(∗, ∗, ∗, ∗)], [M^{∗}(∗, ∗, ∗)], [macro], [s_{0}], [zip(∗, ∗)], [assoc_{1}(∗, ∗, ∗)], [(∗)^{p}], [self], [[∗ ¨= ∗]], [[∗ ˙= ∗]], [[∗ ´= ∗]], [[∗ =^{pyk} ∗]], [[∗ =^{tex} ∗]], [[∗ =^{name} ∗]], [Priority table[∗]], [M̃_{1}], [M̃_{2}(∗)], [M̃_{3}(∗)], [M̃_{4}(∗, ∗, ∗, ∗)], [M̃(∗, ∗, ∗)], [Q̃(∗, ∗, ∗)], [Q̃_{2}(∗, ∗, ∗)], [Q̃_{3}(∗, ∗, ∗, ∗)], [Q̃^{∗}(∗, ∗, ∗)], [(∗)], [(∗)], [display(∗)], [statement(∗)], [[∗]^{·}], [[∗]^{−}], [aspect(∗, ∗)], [aspect(∗, ∗, ∗)], [⟨∗⟩], [tuple_{1}(∗)], [tuple_{2}(∗)], [let_{2}(∗, ∗)], [let_{1}(∗, ∗)], [[∗ =^{claim} ∗]], [checker], [check(∗, ∗)], [check_{2}(∗, ∗, ∗)], [check_{3}(∗, ∗, ∗)], [check^{∗}(∗, ∗)], [check_{2}^{∗}(∗, ∗, ∗)], [[∗]^{·}], [[∗]^{−}], [[∗]^{∘}], [msg], [[∗ =^{msg} ∗]], [<stmt>], [stmt], [[∗ =^{stmt} ∗]], [HeadNil′], [HeadPair′], [Transitivity′], [⊥ ⊥], [Contra′], [T′_{E}], [L_{1}], [\underline{∗}], [A], [B], [C], [D], [E], [F], [G], [H], [I], [J], [K], [L], [M], [N], [O], [P], [Q], [R], [S], [T ], [U], [V], [W], [X], [Y], [Z], [⟨∗ | ∗:= ∗⟩], [⟨^{∗}∗ | ∗:= ∗⟩], [∅], [Remainder], [(∗)^{v}], [intro(∗, ∗, ∗, ∗)], [intro(∗, ∗, ∗)], [error(∗, ∗)], [error_{2}(∗, ∗)], [proof(∗, ∗, ∗)], [proof_{2}(∗, ∗)], [S(∗, ∗)], [S^{I}(∗, ∗)], [S^{⊳} (∗, ∗)], [S_{1}^{⊳} (∗, ∗, ∗)], [S^{E}(∗, ∗)], [S_{1}^{E}(∗, ∗, ∗)], [S^{+}(∗, ∗)], [S_{1}^{+}(∗, ∗, ∗)], [S^{−}(∗, ∗)], [S_{1}^{−}(∗, ∗, ∗)], [S^{∗}(∗, ∗)], [S_{1}^{∗}(∗, ∗, ∗)], [S_{2}^{∗}(∗, ∗, ∗, ∗)], [S^{@}(∗, ∗)],

[S_{1}^{@}(∗, ∗, ∗)], [S^{⊢}(∗, ∗)], [S_{1}^{⊢}(∗, ∗, ∗, ∗)], [S⊢ ⊢(∗, ∗)], [S⊢ ⊢ ; 1(∗, ∗, ∗, ∗)], [S^{i.e.}(∗, ∗)],

[S_{1}^{i.e.}(∗, ∗, ∗, ∗)], [S_{2}^{i.e.}(∗, ∗, ∗, ∗, ∗)], [S^{∀}(∗, ∗)], [S_{1}^{∀}(∗, ∗, ∗, ∗)], [S^{;}(∗, ∗)], [S_{1}^{;}(∗, ∗, ∗)], [S_{2}^{;}(∗, ∗, ∗, ∗)], [T (∗)], [claims(∗, ∗, ∗)], [claims_{2}(∗, ∗, ∗)], [<proof>], [proof], [[Lemma ∗: ∗]], [[Proof of ∗: ∗]], [[∗ lemma ∗: ∗]], [[∗ antilemma ∗: ∗]], [[∗ rule ∗: ∗]], [[∗ antirule ∗: ∗]], [verifier], [V_{1}(∗)], [V_{2}(∗, ∗)], [V_{3}(∗, ∗, ∗, ∗)], [V_{4}(∗, ∗)], [V_{5}(∗, ∗, ∗, ∗)], [V_{6}(∗, ∗, ∗, ∗)], [V_{7}(∗, ∗, ∗, ∗)], [Cut(∗, ∗)], [Head_{⊕}(∗)], [Tail_{⊕}(∗)], [rule_{1}(∗, ∗)], [rule(∗, ∗)], [Rule tactic], [Plus(∗, ∗)], [[Theory ∗]], [theory_{2}(∗, ∗)], [theory_{3}(∗, ∗)], [theory_{4}(∗, ∗, ∗)], [HeadNil′′], [HeadPair′′], [Transitivity′′], [Contra′′], [HeadNil], [HeadPair], [Transitivity], [Contra], [T_{E}], [ragged right], [ragged right expansion ], [parm(∗, ∗, ∗)], [parm^{∗}(∗, ∗, ∗)], [inst(∗, ∗)], [inst^{∗}(∗, ∗)], [occur(∗, ∗, ∗)], [occur^{∗}(∗, ∗, ∗)], [unify(∗ = ∗, ∗)], [unify^{∗}(∗ = ∗, ∗)], [unify_{2}(∗ = ∗, ∗)], [L_{a}], [L_{b}], [L_{c}], [L_{d}], [L_{e}], [L_{f}], [L_{g}], [L_{h}], [L_{i}], [L_{j}], [L_{k}], [L_{l}], [L_{m}], [L_{n}], [L_{o}], [L_{p}], [L_{q}], [L_{r}], [L_{s}], [L_{t}], [L_{u}], [L_{v}], [L_{w}], [L_{x}], [L_{y}], [L_{z}], [L_{A}], [L_{B}], [L_{C}], [L_{D}], [L_{E}], [L_{F}], [L_{G}], [L_{H}], [L_{I}], [L_{J}], [L_{K}], [L_{L}], [L_{M}], [L_{N}], [L_{O}], [L_{P}], [L_{Q}], [L_{R}], [L_{S}], [L_{T}], [L_{U}], [L_{V}], [L_{W}], [L_{X}], [L_{Y}], [L_{Z}], [L_{?}], [Reflexivity], [Reflexivity_{1}], [Commutativity], [Commutativity_{1}], [<tactic>], [tactic], [[∗ =^{tactic} ∗]], [P(∗, ∗, ∗)], [P^{∗}(∗, ∗, ∗)], [p_{0}], [conclude_{1}(∗, ∗)], [conclude_{2}(∗, ∗, ∗)], [conclude_{3}(∗, ∗, ∗, ∗)], [conclude_{4}(∗, ∗)], [check], [[∗ =^{∘} ∗]], [RootVisible(∗)], [A], [R], [C], [T], [L], [\{∗\}], [\overline{∗}], [a], [b], [c], [d], [e], [f ], [g], [h], [i], [j ], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [x], [y], [z], [⟨∗≡ ∗ | ∗:=∗⟩], [⟨∗≡^{0} ∗ | ∗:=∗⟩], [⟨∗≡^{1} ∗ | ∗:=∗⟩], [⟨∗≡^{∗} ∗ | ∗:=∗⟩], [Ded(∗, ∗)], [Ded_{0}(∗, ∗)], [Ded_{1}(∗, ∗, ∗)], [Ded_{2}(∗, ∗, ∗)], [Ded_{3}(∗, ∗, ∗, ∗)], [Ded_{4}(∗, ∗, ∗, ∗)], [Ded_{4}^{∗}(∗, ∗, ∗, ∗)], [Ded_{5}(∗, ∗, ∗)], [Ded_{6}(∗, ∗, ∗, ∗)], [Ded_{6}^{∗}(∗, ∗, ∗, ∗)], [Ded_{7}(∗)], [Ded_{8}(∗, ∗)], [Ded_{8}^{∗}(∗, ∗)], [S], [Neg], [MP], [Gen], [Ded], [S1], [S2], [S3], [S4], [S5], [S6], [S7], [S8], [S9], [Repetition], [A1′], [A2′], [A4′], [A5′], [Prop 3.2a], [Prop 3.2b], [Prop 3.2c], [Prop 3.2d], [Prop 3.2e_{1}], [Prop 3.2e_{2}], [Prop 3.2e], [Prop 3.2f_{1}], [Prop 3.2f_{2}], [Prop 3.2f], [Prop 3.2g_{1}], [Prop 3.2g_{2}], [Prop 3.2g], [Prop 3.2h_{1}], [Prop 3.2h_{2}], [Prop 3.2h], [Block_{1}(∗, ∗, ∗)], [Block_{2}(∗)], [(⋯)], [Objekt-var], [Ex-var], [Ph-var], [Værdi], [Variabel], [Op(∗)], [Op(∗, ∗)], [∗ ¨= ∗], [ContainsEmpty(∗)], [Dedu(∗, ∗)], [Dedu_{0}(∗, ∗)], [Dedu_{s}(∗, ∗, ∗)], [Dedu_{1}(∗, ∗, ∗)], [Dedu_{2}(∗, ∗, ∗)], [Dedu_{3}(∗, ∗, ∗, ∗)], [Dedu_{4}(∗, ∗, ∗, ∗)], [Dedu_{4}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{5}(∗, ∗, ∗)], [Dedu_{6}(∗, ∗, ∗, ∗)], [Dedu_{6}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{7}(∗)], [Dedu_{8}(∗, ∗)], [Dedu_{8}^{∗}(∗, ∗)], [Ex_{1}], [Ex_{2}], [Ex_{10}], [Ex_{20}], [∗_{Ex}], [∗^{Ex}], [⟨∗≡ ∗ | ∗:=∗⟩_{Ex}], [⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ex}], [⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ex}], [⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ex}], [ph_{1}], [ph_{2}], [ph_{3}], [∗_{Ph}], [∗^{Ph}], [⟨∗≡ ∗ | ∗:=∗⟩_{Ph}], [⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ph}], [⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ph}], [⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ph}], [bs], [OBS], [BS], [Ø], [ZFsub], [MP], [Gen], [Repetition], [Neg], [Ded], [ExistIntro], [Extensionality], [Ødef], [PairDef], [UnionDef], [PowerDef], [SeparationDef], [CheatAllDisjoint], [AddDoubleNeg], [RemoveDoubleNeg], [AndCommutativity], [AutoImply], [Contrapositive], [FirstConjunct], [SecondConjunct], [FromContradiction], [FromDisjuncts], [IffCommutativity], [IffFirst], [IffSecond], [ImplyTransitivity], [JoinConjuncts], [MP2], [MP3], [MP4], [MP5], [MT], [NegativeMT], [Technicality], [Weakening], [WeakenOr1], [WeakenOr2], [Formula2Pair], [Pair2Formula], [Formula2Union], [Union2Formula], [Formula2Sep], [Sep2Formula], [SubsetInPower], [HelperPowerIsSub], [PowerIsSub], [(Switch)HelperPowerIsSub], [(Switch)PowerIsSub], [ToSetEquality], [HelperToSetEquality(t)], [ToSetEquality(t)], [HelperFromSetEquality], [FromSetEquality], [HelperReflexivity], [Reflexivity], [HelperSymmetry], [Symmetry], [HelperTransitivity], [Transitivity], [ERisReflexive], [ERisSymmetric], [ERisTransitive], [ØisSubset], [HelperMemberNotØ], [MemberNotØ], [HelperUniqueØ], [UniqueØ], [=Reflexivity], [=Symmetry], [Helper=Transitivity], [=Transitivity], [HelperTransferNotEq], [TransferNotEq], [HelperPairSubset], [Helper(2)PairSubset], [PairSubset], [SamePair], [SameSingleton], [UnionSubset], [SameUnion], [SeparationSubset], [SameSeparation], [SameBinaryUnion], [IntersectionSubset], [SameIntersection], [AutoMember], [HelperEqSysNotØ], [EqSysNotØ], [HelperEqSubset], [EqSubset], [HelperEqNecessary], [EqNecessary], [HelperNoneEqNecessary], [Helper(2)NoneEqNecessary], [NoneEqNecessary], [EqClassIsSubset], [EqClassesAreDisjoint], [AllDisjoint], [AllDisjointImply], [BSsubset], [Union(BS/R)subset], [UnionIdentity], [EqSysIsPartition]; Preassociative [∗ {∗}], [∗/indexintro(∗, ∗, ∗, ∗)], [∗/intro(∗, ∗, ∗)], [∗/bothintro(∗, ∗, ∗, ∗, ∗)], [∗/nameintro(∗, ∗, ∗, ∗)], [∗′], [∗[ ∗]], [∗[∗→∗]], [∗[∗⇒∗]], [∗0], [∗1], [0b], [∗-color(∗)], [∗-color^{∗}(∗)], [∗^{H}], [∗^{T}], [∗^{U}], [∗^{h}], [∗^{t}], [∗^{s}], [∗^{c}], [∗^{d}], [∗^{a}], [∗^{C}], [∗^{M}], [∗^{B}], [∗^{r}], [∗^{i}], [∗^{d}], [∗^{R}], [∗^{0}], [∗^{1}], [∗^{2}], [∗^{3}], [∗^{4}], [∗^{5}], [∗^{6}], [∗^{7}], [∗^{8}], [∗^{9}], [∗^{E}], [∗^{V}], [∗^{C}], [∗^{C^{∗}}], [∗^{hide}]; Preassociative [“ ∗ ”], [], [(∗)^{t}], [string(∗) + ∗], [string(∗) ++ ∗], [ ∗], [ ∗], [!∗], ["∗], [#∗], [$∗], [%∗], [&∗], [’∗], [(∗], [)∗], [∗∗], [+∗], [, ∗], [-∗], [.∗], [/∗], [0∗], [1∗], [2∗], [3∗], [4∗], [5∗], [6∗], [7∗], [8∗], [9∗], [:∗], [; ∗], [<∗], [=∗], [>∗], [?∗], [@∗], [A∗], [B∗], [C∗], [D∗], [E∗], [F∗], [G∗], [H∗], [I∗], [J∗], [K∗], [L∗], [M∗], [N∗], [O∗], [P∗], [Q∗], [R∗], [S∗], [T∗], [U∗], [V∗], [W∗], [X∗], [Y∗], [Z∗], [[∗], [\∗], []∗], [ˆ ∗], [ ∗], [‘∗], [a∗], [b∗], [c∗], [d∗], [e∗], [f∗], [g∗], [h∗], [i∗], [j∗], [k∗], [l∗], [m∗], [n∗], [o∗], [p∗], [q∗], [r∗], [s∗], [t∗], [u∗], [v∗], [w∗], [x∗], [y∗], [z∗], [{∗], [|∗], [}∗], [˜ ∗], [Preassociative ∗; ∗], [Postassociative ∗; ∗], [[∗], ∗], [priority ∗ end], [newline ∗], [macro newline ∗], [MacroIndent(∗)]; Preassociative [∗ ’ ∗], [∗ ‘ ∗]; Preassociative [∗′]; Preassociative [∗/∗], [∗ ∩ ∗]; Preassociative [∪∗], [∗ ∪ ∗], [P(∗)]; Preassociative [{∗}]; Preassociative [{∗, ∗}], [⟨∗, ∗⟩]; Preassociative [∗ ∈ ∗], [∗(∗, ∗)], [ReflRel(∗, ∗)], [SymRel(∗, ∗)], [TransRel(∗, ∗)], [EqRel(∗, ∗)], [[∗ ∈ ∗]_{∗}], [Partition(∗, ∗)]; Preassociative [∗ · ∗], [∗ ·_{0} ∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗]; Preassociative [∗ ∪ {∗}], [∗ ∪ ∗], [∗\{∗}]; Postassociative [∗ ˙. . ∗], [∗ \underline{˙. .} ∗], [∗ \underline{: :} ∗], [∗ \underline{+2∗} ∗], [∗ : : ∗], [∗ +2∗ ∗]; Postassociative [∗, ∗]; Preassociative

[∗ ≈^{B} ∗], [∗ ≈^{D} ∗], [∗ ≈^{C} ∗], [∗ ≈^{P} ∗], [∗ ≈ ∗], [∗ = ∗], [∗ →^{+} ∗], [∗ =^{t} ∗], [∗ =^{t^{∗}} ∗], [∗ =^{r} ∗],

[∗ ∈_{t} ∗], [∗ ⊆_{T} ∗], [∗ =^{T} ∗], [∗ =^{s} ∗], [∗ free in ∗], [∗ free in^{∗} ∗], [∗ free for ∗ in ∗], [∗ free for^{∗} ∗ in ∗], [∗ ∈_{c} ∗], [∗ < ∗], [∗ <′ ∗], [∗ ≤′ ∗], [∗ = ∗], [∗ ≠ ∗], [∗^{var}], [∗#^{0}∗], [∗#^{1}∗], [∗#^{∗}∗], [∗=∗], [∗ ⊆ ∗]; Preassociative [¬∗], [ ˙¬ ∗], [∗ ∉ ∗], [∗ ≠ ∗]; Preassociative [∗ ∧ ∗], [∗ ¨∧ ∗], [∗ ˜∧ ∗], [∗ ∧_{c} ∗], [∗ ˙∧ ∗]; Preassociative [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗], [∗ ˙∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗], [∗ ˙⇔ ∗]; Preassociative [{ph ∈ ∗ | ∗}]; Postassociative [∗ : ∗], [∗ spy ∗], [∗!∗]; Preassociative

[∗ \{ \begin{matrix}∗ \\ ∗ \end{matrix} ];

Preassociative [λ ∗ .∗], [Λ ∗ .∗], [Λ∗], [if ∗ then ∗ else ∗], [let ∗ = ∗ in ∗], [let ∗ ¨= ∗ in ∗]; Preassociative [∗#∗]; Preassociative [∗^{I}], [∗^{⊳} ], [∗^{V}], [∗^{+}], [∗^{−}], [∗^{∗}]; Preassociative [∗ @ ∗], [∗ ⊳ ∗], [∗ ⊳ ⊳ ∗], [∗ ≫ ∗], [∗ ⊵ ∗]; Postassociative [∗ ⊢ ∗], [∗ ⊢⊢ ∗], [∗ i.e. ∗]; Preassociative [∀∗: ∗], [Π∗: ∗]; Postassociative [∗ ⊕ ∗]; Postassociative [∗; ∗]; Preassociative [∗ proves ∗]; Preassociative [∗ proof of ∗ : ∗], [Line ∗ : ∗ ≫ ∗; ∗], [Last line ∗ ≫ ∗ □], [Line ∗ : Premise ≫ ∗; ∗], [Line ∗ : Side-condition ≫ ∗; ∗], [Arbitrary ≫ ∗; ∗], [Local ≫ ∗ = ∗; ∗], [Begin ∗; ∗ : End; ∗], [Last block line ∗ ≫ ∗; ], [Arbitrary ≫ ∗; ∗]; Postassociative [∗ | ∗]; Postassociative [∗ , ∗], [∗[ ∗]∗]; Preassociative [∗&∗]; Preassociative [∗\\∗], [∗ linebreak[4] ∗], [∗\\∗]; ]

[EquivalenceRelations →^{tex} “EquivalenceRelations”]

[EquivalenceRelations →^{pyk} “equivalence-relations”]

(⋯)

[(⋯) →^{tex} “(\cdots{})”]

[(⋯) →^{pyk} “cdots”]

Objekt-var

[Objekt-var →^{tex} “\texttt{Objekt-var}”]

[Objekt-var →^{pyk} “object-var”]

Ex-var

[Ex-var →^{tex} “\texttt{Ex-var}”]

[Ex-var →^{pyk} “ex-var”]

Ph-var

[Ph-var →^{tex} “\texttt{Ph-var}”]

[Ph-var →^{pyk} “ph-var”]

Værdi

[Værdi →^{tex} “\texttt{V\ae{}rdi}”]

[Værdi →^{pyk} “vaerdi”]

Variabel

[Variabel →^{tex} “\texttt{Variabel}”]

[Variabel →^{pyk} “variabel”]

Op(∗)

[Op(x) →^{tex} “Op(#1. )”]

[Op(∗) →^{pyk} “op " end op”]

Op(∗, ∗)

[Op(x, y) →^{tex} “Op(#1. , #2. )”]

[Op(∗, ∗) →^{pyk} “op2 " comma " end op2”]

∗ ¨= ∗

[x ¨= y →^{tex} “#1.

\mathrel {\ddot {=}} #2.”]

[∗ ¨= ∗ →^{pyk} “define-equal " comma " end equal”]

ContainsEmpty(∗)

[ContainsEmpty(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ContainsEmpty(x) ¨= {ph ∈ x | Ø ∈ ph_{1}}]⌉)]

[ContainsEmpty(x) →^{tex} “ContainsEmpty(#1. )”]

[ContainsEmpty(∗) →^{pyk} “contains-empty " end empty”]

Dedu(∗, ∗)

[Dedu(p, c) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Dedu(p, c) ¨= λx.Dedu_{0}(⌈p⌉, ⌈c⌉)]⌉)]

[Dedu(x, y) →^{tex} “ Dedu(#1. , #2. )”]

[Dedu(∗, ∗) →^{pyk} “1deduction " conclude " end 1deduction”]

Dedu_{0}(∗, ∗)

[Dedu_{0}(p, c) →^{val} c!If(Dedu_{8}(p, T), Dedu_{s}(Dedu_{7}(p), c, T), F)]

[Dedu_{0}(x, y) →^{tex} “ Dedu 0(#1. , #2. )”]

[Dedu_{0}(∗, ∗) →^{pyk} “1deduction zero " conclude " end 1deduction”]

Dedu_{s}(∗, ∗, ∗)

[Dedu_{s}(p, c, s) →^{val} If(p =^{r} ⌈x ⊢⊢ y⌉, c =^{r} ⌈x ⊢⊢ y⌉ ∧ p^{1} =^{t} c^{1} ∧ Dedu_{s}(p^{2}, c^{2}, c^{1} : : s), Dedu_{1}(p, c, s))]

[Dedu_{s}(x, y, z) →^{tex} “Dedu {s}(#1. , #2. , #3. )”]

[Dedu_{s}(∗, ∗, ∗) →^{pyk} “1deduction side " conclude " condition " end 1deduction”]

Dedu_{1}(∗, ∗, ∗)

[Dedu_{1}(p, c, s) →^{val} If(c =^{r} ⌈x ⊢⊢ y⌉, Dedu_{1}(p, c^{2}, c^{1} : : s), Dedu_{2}(p, c, s))]

[Dedu_{1}(x, y, z) →^{tex} “ Dedu 1(#1. , #2. , #3. )”] [Dedu_{1}(∗, ∗, ∗) →^{pyk} “1deduction one " conclude " condition " end 1deduction”]

Dedu_{2}(∗, ∗, ∗)

[Dedu_{2}(p, c, s) →^{val} s!p =^{r} ⌈x ⊢ y⌉ ∧ c =^{r} ⌈x ⇒

y⌉ \{ \begin{matrix}Dedu_{3}(p^{1}, c^{1}, s, T) ∧ Dedu_{2}(p^{2}, c^{2}, s) \\ Dedu_{4}(p, c, s, Dedu_{6}(p, c, T, T)) \end{matrix} ]

[Dedu_{2}(x, y, z) →^{tex} “ Dedu 2(#1. , #2. , #3. )”]

[Dedu_{2}(∗, ∗, ∗) →^{pyk} “1deduction two " conclude " condition " end 1deduction”]

Dedu_{3}(∗, ∗, ∗, ∗)

[Dedu_{3}(p, c, s, b) →^{val} If(¬c =^{r} ⌈∀_{obj}x: y⌉, Dedu_{4}(p, c, s, b), If(p =^{r} ⌈∀_{obj}x: y⌉ ∧ p^{1} =^{t} c^{1}, Dedu_{4}(p, c, s, b), Dedu_{3}(p, c^{2}, s, c^{1} : : c^{1} : : b)))]

[Dedu_{3}(x, y, z, u) →^{tex} “ Dedu 3(#1. , #2. , #3. , #4. )”]

[Dedu_{3}(∗, ∗, ∗, ∗) →^{pyk} “1deduction three " conclude " condition " bound " end 1deduction”]

Dedu_{4}(∗, ∗, ∗, ∗)

[Dedu_{4}(p, c, s, b) →^{val} s!b!If(p =^{r} ⌈\overline{x}⌉, lookup(p, b, T) =^{t} c, If(¬p =^{r} c, F, If(p =^{r} ⌈∀_{obj}x: y⌉, p^{1} =^{t} c^{1} ∧ Dedu_{4}(p^{2}, c^{2}, s, p^{1} : : p^{1} : : b), If(¬p =^{r} ⌈\underline{x}⌉, Dedu_{4}^{∗}(p^{t}, c^{t}, s, b), p^{1} =^{t} c^{1} ∧ Dedu_{5}(p, s, b)))))]

[Dedu_{4}(x, y, z, u) →^{tex} “ Dedu 4(#1. , #2. , #3.

, #4. )”]

[Dedu_{4}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four " conclude " condition " bound " end 1deduction”]

Dedu_{4}^{∗}(∗, ∗, ∗, ∗)

[Dedu_{4}^{∗}(p, c, s, b) →^{val} c!s!b!If(p, T, Dedu_{4}(p^{h}, c^{h}, s, b) ∧ Dedu_{4}^{∗}(p^{t}, c^{t}, s, b))]

[Dedu_{4}^{∗}(x, y, z, u) →^{tex} “ Dedu 4ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{4}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four star " conclude " condition " bound " end 1deduction”]

Dedu_{5}(∗, ∗, ∗)

[Dedu_{5}(p, s, b) →^{val} p!s!If(b, T, ⌈⌈x⌉#^{0}⌈y⌉⌉^{h} : : ⌈⌈∗⌉⌉^{h} : : b^{h}^{h} : : T : : ⌈⌈x⌉⌉^{h} : : p : : T : : T ∈_{t} s ∧ Dedu_{5}(p, s, b^{t}))]

[Dedu_{5}(x, y, z) →^{tex} “ Dedu 5(#1. , #2. , #3. )”]

[Dedu_{5}(∗, ∗, ∗) →^{pyk} “1deduction five " condition " bound " end 1deduction”]

Dedu_{6}(∗, ∗, ∗, ∗)

[Dedu_{6}(p, c, e, b) →^{val} p!c!b!e!If(p =^{r} ⌈\overline{x}⌉, p ∈_{t} e \{ \begin{matrix}b \\ p : : c : : b \end{matrix} , If(¬p =^{r} c, T,

If(p =^{r} ⌈\underline{a}⌉, b, If(p =^{r} ⌈∀_{obj}x: y⌉, Dedu_{6}(p^{2}, c^{2}, c^{1} : : e, b), Dedu_{6}^{∗}(p^{t}, c^{t}, e, b)))))]

[Dedu_{6}(p, c, e, b) →^{tex} “ Dedu 6(#1. , #2. , #3.

[Dedu_{6}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six " conclude " exception " bound " end 1deduction”]

Dedu_{6}^{∗}(∗, ∗, ∗, ∗)

[Dedu_{6}^{∗}(p, c, e, b) →^{val} p!c!b!e!If(p, b, Dedu_{6}^{∗}(p^{t}, c^{t}, e, Dedu_{6}(p^{h}, c^{h}, e, b)))]

[Dedu_{6}^{∗}(p, c, e, b) →^{tex} “ Dedu 6ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{6}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six star " conclude " exception " bound " end 1deduction”]

Dedu_{7}(∗)

[Dedu_{7}(p) →^{val} p =^{r} ⌈∀x: y⌉ \{ \begin{matrix}Dedu_{7}(p^{2}) \\ p \end{matrix} ]

[Dedu_{7}(p) →^{tex} “ Dedu 7(#1. )”]

[Dedu_{7}(∗) →^{pyk} “1deduction seven " end 1deduction”]

Dedu_{8}(∗, ∗)

[Dedu_{8}(p, b) →^{val} If(p =^{r} ⌈∀x: y⌉, Dedu_{8}(p^{2}, p^{1} : : b), If(p =^{r} ⌈\underline{a}⌉, p ∈_{t} b, Dedu_{8}^{∗}(p^{t}, b)))]

[Dedu_{8}(p, b) →^{tex} “ Dedu 8(#1. , #2. )”]

[Dedu_{8}(∗, ∗) →^{pyk} “1deduction eight " bound " end 1deduction”]

Dedu_{8}^{∗}(∗, ∗)

[Dedu_{8}^{∗}(p, b) →^{val} b!If(p, T, If(Dedu_{8}(p^{h}, b), Dedu_{8}^{∗}(p^{t}, b), F))] [Dedu_{8}^{∗}(p, b) →^{tex} “ Dedu 8ˆ ∗(#1. , #2. )”] [Dedu_{8}^{∗}(∗, ∗) →^{pyk} “1deduction eight star " bound " end 1deduction”]

Ex_{1}

[Ex_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{1} ¨= a_{Ex}]⌉)]

[Ex_{1} →^{tex} “Ex {1}”]

[Ex_{1} →^{pyk} “ex1”]

Ex_{2}

[Ex_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{2} ¨= b_{Ex}]⌉)]

[Ex_{2} →^{tex} “Ex {2}”]

[Ex_{2} →^{pyk} “ex2”]

Ex_{10}

[Ex_{10} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{10} ¨= j_{Ex}]⌉)]

[Ex_{10} →^{tex} “Ex {10}”]

[Ex_{10} →^{pyk} “ex10”]

Ex_{20}

[Ex_{20} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{20} ¨= t_{Ex}]⌉)]

[Ex_{20} →^{tex} “Ex {20}”]

[Ex_{20} →^{pyk} “ex20”]

∗_{Ex}

[x_{Ex} →^{tex} “#1.

{Ex}”]

[∗_{Ex} →^{pyk} “existential var " end var”]

∗^{Ex}

[x^{Ex} →^{val} x =^{r} ⌈x_{Ex}⌉]

[x^{Ex} →^{tex} “#1.

ˆ {Ex}”]

[∗^{Ex} →^{pyk} “" is existential var”]

⟨∗≡ ∗ | ∗:=∗⟩_{Ex}

[⟨a≡b|x:=t⟩_{Ex} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨a≡b|x:=t⟩_{Ex} ¨= ⟨⌈a⌉≡^{0}⌈b⌉|⌈x⌉:=⌈t⌉⟩_{Ex}]⌉)]

[⟨x≡y|z:=u⟩_{Ex} →^{tex} “\langle #1. {\equiv} #2. | #3. {:=} #4. \rangle {Ex} ”]

[⟨∗≡ ∗ | ∗:=∗⟩_{Ex} →^{pyk} “exist-sub " is " where " is " end sub”]

⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ex}

[⟨a≡^{0}b|x:=t⟩_{Ex} →^{val} λc.x^{Ex} ∧ ⟨a≡^{1}b|x:=t⟩_{Ex}]

[⟨x≡^{0}y|z:=u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 0 #2. | #3. {:=} #4. \rangle {Ex} ”]

[⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ex} →^{pyk} “exist-sub0 " is " where " is " end sub”]

⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ex}

[⟨a≡^{1}b|x:=t⟩_{Ex} →^{val} a!x!t! If(b =^{r} ⌈∀_{obj}u: v⌉, F, If(b^{Ex} ∧ b =^{t} x, a =^{t} t, If( a =^{r} b, ⟨a^{t}≡^{∗}b^{t}|x:=t⟩_{Ex}, F)))]

[⟨x≡^{1}y|z:=u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:=} #4. \rangle {Ex} ”]

[⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ex} →^{pyk} “exist-sub1 " is " where " is " end sub”]

⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ex}

[⟨a≡^{∗}b|x:=t⟩_{Ex} →^{val} b!x!t!If(a, T, If(⟨a^{h}≡^{1}b^{h}|x:=t⟩_{Ex}, ⟨a^{t}≡^{∗}b^{t}|x:=t⟩_{Ex}, F))]

[⟨x≡^{∗}y|z:=u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:=} #4. \rangle {Ex} ”]

[⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ex} →^{pyk} “exist-sub∗ " is " where " is " end sub”]

ph_{1}

[ph_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{1} ¨= a_{Ph}]⌉)]

[ph_{1} →^{tex} “ph {1}”]

[ph_{1} →^{pyk} “placeholder-var1”]

ph_{2}

[ph_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{2} ¨= b_{Ph}]⌉)]

[ph_{2} →^{tex} “ph {2}”]

[ph_{2} →^{pyk} “placeholder-var2”]

ph_{3}

[ph_{3} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{3} ¨= c_{Ph}]⌉)]

[ph_{3} →^{tex} “ph {3}”]

[ph_{3} →^{pyk} “placeholder-var3”]

∗_{Ph}

[x_{Ph} →^{tex} “#1.

{Ph} ”]

[∗_{Ph} →^{pyk} “placeholder-var " end var”]

∗^{Ph}

[x^{Ph} →^{val} x =^{r} ⌈x_{Ph}⌉]

[x^{Ph} →^{tex} “#1.

ˆ {Ph}”]

[∗^{Ph} →^{pyk} “" is placeholder-var”]

⟨∗≡ ∗ | ∗:=∗⟩_{Ph}

[⟨a≡b|x:=t⟩_{Ph} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨a≡b|x:=t⟩_{Ph} ¨= ⟨⌈a⌉≡^{0}⌈b⌉|⌈x⌉:=⌈t⌉⟩_{Ph}]⌉)]

[⟨x≡y|z:=u⟩_{Ph} →^{tex} “\langle #1. {\equiv} #2. | #3. {:=} #4. \rangle {Ph} ”]

[⟨∗≡ ∗ | ∗:=∗⟩_{Ph} →^{pyk} “ph-sub " is " where " is " end sub”]

⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ph}

[⟨a≡^{0}b|x:=t⟩_{Ph} →^{val} λc.x^{Ph} ∧ ⟨a≡^{1}b|x:=t⟩_{Ph}]

[⟨x≡^{0}y|z:=u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 0 #2.

| #3. {:=} #4. \rangle {Ph} ”]

[⟨∗≡^{0} ∗ | ∗:=∗⟩_{Ph} →^{pyk} “ph-sub0 " is " where " is " end sub”]

⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ph}

[⟨a≡^{1}b|x:=t⟩_{Ph} →^{val} a!x!t! If(b =^{r} ⌈∀_{obj}u: v⌉, F, If(b^{Ph} ∧ b =^{t} x, a =^{t} t, If(b^{Ex}, a =^{r} b, If( a =^{r} b, ⟨a^{t}≡^{∗}b^{t}|x:=t⟩_{Ph}, F))))]

[⟨x≡^{1}y|z:=u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:=} #4. \rangle {Ph} ”]

[⟨∗≡^{1} ∗ | ∗:=∗⟩_{Ph} →^{pyk} “ph-sub1 " is " where " is " end sub”]

⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ph}

[⟨a≡^{∗}b|x:=t⟩_{Ph} →^{val} b!x!t!If(a, T, If(⟨a^{h}≡^{1}b^{h}|x:=t⟩_{Ph}, ⟨a^{t}≡^{∗}b^{t}|x:=t⟩_{Ph}, F))]

[⟨x≡^{∗}y|z:=u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:=} #4. \rangle {Ph} ”]

[⟨∗≡^{∗} ∗ | ∗:=∗⟩_{Ph} →^{pyk} “ph-sub∗ " is " where " is " end sub”]

bs

[bs →^{tex} “\mathsf {bs}”]

[bs →^{pyk} “var big set”]

OBS

[OBS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[OBS ¨= \overline{bs}]⌉)]

[OBS →^{tex} “ \mathsf {OBS}”]

[OBS →^{pyk} “object big set”]

BS

[BS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[BS ¨= \underline{bs}]⌉)]

[BS →^{tex} “{\cal BS}”]

[BS →^{pyk} “meta big set”]

Ø

[Ø →^{tex} “\mathrm{\O}”]

[Ø →^{pyk} “zermelo empty set”]

ZFsub

[ZFsub →^{stmt} ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}) ⊕ ∀\underline{a}: \underline{a} ⊢ \underline{a} ⊕ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}= b_{Ph}\} ⊢ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ ˙¬ \underline{x}=\underline{y} ⊢ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø ⊕ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊕ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:=⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x} ⊕ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a} ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{z}⌉⟩_{Ph} ⊢⊢ ˙¬ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊕ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{s}: ˙¬ \underline{s} ∈ Ø]

[ZFsub →^{tex} “ZFsub”]

[ZFsub →^{pyk} “system zf”]

MP

[MP →^{proof} Rule tactic] [MP →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b}]

[MP →^{tex} “MP”]

[MP →^{pyk} “1rule mp”]

Gen

[Gen →^{proof} Rule tactic] [Gen →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a}] [Gen →^{tex} “Gen”] [Gen →^{pyk} “1rule gen”]

Repetition

[Repetition →^{proof} Rule tactic] [Repetition →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ \underline{a}] [Repetition →^{tex} “Repetition”] [Repetition →^{pyk} “1rule repetition”]

Neg

[Neg →^{proof} Rule tactic] [Neg →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{b}] [Neg →^{tex} “Neg”] [Neg →^{pyk} “1rule ad absurdum”]

Ded

[Ded →^{proof} Rule tactic] [Ded →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b}] [Ded →^{tex} “Ded”] [Ded →^{pyk} “1rule deduction”]

ExistIntro

[ExistIntro →^{proof} Rule tactic] [ExistIntro →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:=⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b}] [ExistIntro →^{tex} “ExistIntro”] [ExistIntro →^{pyk} “1rule exist intro”]

Extensionality

[Extensionality →^{proof} Rule tactic] [Extensionality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y}] [Extensionality →^{tex} “Extensionality”] [Extensionality →^{pyk} “axiom extensionality”]

Ødef

[Ødef →^{proof} Rule tactic] [Ødef →^{stmt} ZFsub ⊢ ∀\underline{s}: ˙¬ \underline{s} ∈ Ø] [Ødef →^{tex} “\O{}def”] [Ødef →^{pyk} “axiom empty set”]

PairDef

[PairDef →^{proof} Rule tactic] [PairDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}= \underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\}] [PairDef →^{tex} “PairDef”] [PairDef →^{pyk} “axiom pair definition”]

UnionDef

[UnionDef →^{proof} Rule tactic]

[UnionDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x}]

[UnionDef →^{tex} “UnionDef”]

[UnionDef →^{pyk} “axiom union definition”]

PowerDef

[PowerDef →^{proof} Rule tactic]

[PowerDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x})]

[PowerDef →^{tex} “PowerDef”]

[PowerDef →^{pyk} “axiom power definition”]

SeparationDef

[SeparationDef →^{proof} Rule tactic]

[SeparationDef →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{z}⌉⟩_{Ph} ⊢⊢ ˙¬ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\}]

[SeparationDef →^{tex} “SeparationDef”]

[SeparationDef →^{pyk} “axiom separation definition”]

CheatAllDisjoint

[CheatAllDisjoint →^{proof} Rule tactic]

[CheatAllDisjoint →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ ˙¬ \underline{x}= \underline{y} ⊢ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø] [CheatAllDisjoint →^{tex} “CheatAllDisjoint”]

[CheatAllDisjoint →^{pyk} “cheating axiom all disjoint”]

AddDoubleNeg

[AddDoubleNeg →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ ˙¬ \underline{a} ⊢ RemoveDoubleNeg ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{a}; ∀\underline{a}: Ded⊳∀\underline{a}: ˙¬ ˙¬ ˙¬ \underline{a} ⊢ ˙¬ \underline{a} ≫ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; \underline{a} ⊢ Weakening ⊳\underline{a} ≫ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ \underline{a}; Neg ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ \underline{a} ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ≫ ˙¬ ˙¬ \underline{a}⌉, p_{0}, c)]

[AddDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ ˙¬ ˙¬ \underline{a}]

[AddDoubleNeg →^{tex} “AddDoubleNeg”]

[AddDoubleNeg →^{pyk} “prop lemma add double neg”]

RemoveDoubleNeg

[RemoveDoubleNeg →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ \underline{a} ⊢ Weakening ⊳ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{a} ⇒ ˙¬ ˙¬ \underline{a}; AutoImply ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; Neg⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}⊳ ˙¬ \underline{a} ⇒ ˙¬ ˙¬ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[RemoveDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ \underline{a} ⊢ \underline{a}]

[RemoveDoubleNeg →^{tex} “RemoveDoubleNeg”]

[RemoveDoubleNeg →^{pyk} “prop lemma remove double neg”]

AndCommutativity

[AndCommutativity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{a} ⊢ AddDoubleNeg ⊳ \underline{a} ≫ ˙¬ ˙¬ \underline{a}; MT ⊳ \underline{b} ⇒ ˙¬ \underline{a} ⊳ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{a} ⊢ ˙¬ \underline{b} ≫ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b}; ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ Repetition ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; MT ⊳ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; Repetition ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}⌉, p_{0}, c)]

[AndCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

[AndCommutativity →^{tex} “AndCommutativity”]

[AndCommutativity →^{pyk} “prop lemma and commutativity”]

AutoImply

[AutoImply →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ Repetition ⊳ \underline{a} ≫ \underline{a}; ∀\underline{a}: Ded ⊳ ∀\underline{a}: \underline{a} ⊢ \underline{a} ≫ \underline{a} ⇒ \underline{a}⌉, p_{0}, c)]

[AutoImply →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⇒ \underline{a}]

[AutoImply →^{tex} “AutoImply”]

[AutoImply →^{pyk} “prop lemma auto imply”]

Contrapositive

[Contrapositive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ MT ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a}; ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}⌉, p_{0}, c)]

[Contrapositive →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

[Contrapositive →^{tex} “Contrapositive”]

[Contrapositive →^{pyk} “prop lemma contrapositive”]

FirstConjunct

[FirstConjunct →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ AndCommutativity ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; SecondConjunct ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[FirstConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{a}]

[FirstConjunct →^{tex} “FirstConjunct”]

[FirstConjunct →^{pyk} “prop lemma first conjunct”]

SecondConjunct

[SecondConjunct →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⊢ Weakening ⊳ ˙¬ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{b}; ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; NegativeMT ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ \underline{b}⌉, p_{0}, c)]

[SecondConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{b}]

[SecondConjunct →^{tex} “SecondConjunct”] [SecondConjunct →^{pyk} “prop lemma second conjunct”]

FromContradiction

[FromContradiction →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ Weakening ⊳ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a}; Weakening ⊳ ˙¬ \underline{a} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; Neg ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ \underline{b}⌉, p_{0}, c)]

[FromContradiction →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ \underline{b}]

[FromContradiction →^{tex} “FromContradiction”]

[FromContradiction →^{pyk} “prop lemma from contradiction”]

FromDisjuncts

[FromDisjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Contrapositive ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a}; Technicality ⊳ \underline{a} ⇒ \underline{c} ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{c}; ImplyTransitivity ⊳ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{c} ≫ ˙¬ \underline{b} ⇒ \underline{c}; Contrapositive ⊳ ˙¬ \underline{b} ⇒ \underline{c} ≫ ˙¬ \underline{c} ⇒ ˙¬ ˙¬ \underline{b}; Contrapositive ⊳ \underline{b} ⇒ \underline{c} ≫ ˙¬ \underline{c} ⇒ ˙¬ \underline{b}; Neg ⊳ ˙¬ \underline{c} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{c} ⇒ ˙¬ ˙¬ \underline{b} ≫ \underline{c}⌉, p_{0}, c)]

[FromDisjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{c}]

[FromDisjuncts →^{tex} “FromDisjuncts”]

[FromDisjuncts →^{pyk} “prop lemma from disjuncts”]

IffCommutativity

[IffCommutativity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a}; AndCommutativity ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[IffCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}]

[IffCommutativity →^{tex} “IffCommutativity”]

[IffCommutativity →^{pyk} “prop lemma iff commutativity”]

IffFirst

[IffFirst →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{b} ⊢ SecondConjunct ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ \underline{b} ⇒ \underline{a}; MP ⊳ \underline{b} ⇒ \underline{a} ⊳ \underline{b} ≫ \underline{a}⌉, p_{0}, c)]

[IffFirst →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{b} ⊢ \underline{a}]

[IffFirst →^{tex} “IffFirst”]

[IffFirst →^{pyk} “prop lemma iff first”]

IffSecond

[IffSecond →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{a} ⊢ FirstConjunct ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ \underline{a} ⇒ \underline{b}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}⌉, p_{0}, c)]

[IffSecond →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{a} ⊢ \underline{b}]

[IffSecond →^{tex} “IffSecond”]

[IffSecond →^{pyk} “prop lemma iff second”]

ImplyTransitivity

[ImplyTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{c} ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c}; MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c} ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ⇒ \underline{c} ≫ \underline{a} ⇒ \underline{c}⌉, p_{0}, c)]

[ImplyTransitivity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{c}]

[ImplyTransitivity →^{tex} “ImplyTransitivity”]

[ImplyTransitivity →^{pyk} “prop lemma imply transitivity”]

JoinConjuncts

[JoinConjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ \underline{b} ⊳ \underline{a} ≫ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ⊢ ˙¬ \underline{b} ≫ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b}; \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b} ⊳ \underline{a} ≫ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b}; AddDoubleNeg ⊳ \underline{b} ≫ ˙¬ ˙¬ \underline{b}; MT ⊳ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b} ⊳ ˙¬ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}⌉, p_{0}, c)]

[JoinConjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}] [JoinConjuncts →^{tex} “JoinConjuncts”]

[JoinConjuncts →^{pyk} “prop lemma join conjuncts”]

MP2

[MP2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ≫ \underline{b} ⇒ \underline{c}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}⌉, p_{0}, c)]

[MP2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c}]

[MP2 →^{tex} “MP2”]

[MP2 →^{pyk} “prop lemma mp2”]

MP3

[MP3 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d}; MP ⊳ \underline{c} ⇒ \underline{d} ⊳ \underline{c} ≫ \underline{d}⌉, p_{0}, c)]

[MP3 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d}]

[MP3 →^{tex} “MP3”]

[MP3 →^{pyk} “prop lemma mp3”]

MP4

[MP4 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d} ⇒ \underline{e}; MP2 ⊳ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{c} ⊳ \underline{d} ≫ \underline{e}⌉, p_{0}, c)]

[MP4 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e}]

[MP4 →^{tex} “MP4”]

[MP4 →^{pyk} “prop lemma mp4”]

MP5

[MP5 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ MP3 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{a} ⊳ \underline{b} ⊳ \underline{c} ≫ \underline{d} ⇒ \underline{e} ⇒ \underline{f}; MP2 ⊳ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{d} ⊳ \underline{e} ≫ \underline{f}⌉, p_{0}, c)] [MP5 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ \underline{f}]

[MP5 →^{tex} “MP5”]

[MP5 →^{pyk} “prop lemma mp5”]

MT

[MT →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Technicality ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; NegativeMT ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a}⌉, p_{0}, c)] [MT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a}]

[MT →^{tex} “MT”]

[MT →^{pyk} “prop lemma mt”]

NegativeMT

[NegativeMT →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Weakening ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; Neg ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ \underline{a}⌉, p_{0}, c)] [NegativeMT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ \underline{a}] [NegativeMT →^{tex} “NegativeMT”] [NegativeMT →^{pyk} “prop lemma negative mt”]

Technicality

[Technicality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⊢ RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{a} ≫ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⇒ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] [Technicality →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}] [Technicality →^{tex} “Technicality”] [Technicality →^{pyk} “prop lemma technicality”]

Weakening

[Weakening →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ Repetition ⊳ \underline{b} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ \underline{b} ≫ \underline{b} ⇒ \underline{a} ⇒ \underline{b}; \underline{b} ⊢ MP ⊳ \underline{b} ⇒ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ≫ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[Weakening →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⇒ \underline{b}]

[Weakening →^{tex} “Weakening”]

[Weakening →^{pyk} “prop lemma weakening”]

WeakenOr1

[WeakenOr1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ Weakening ⊳ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[WeakenOr1 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b}]

[WeakenOr1 →^{tex} “WeakenOr1”]

[WeakenOr1 →^{pyk} “prop lemma weaken or first”]

WeakenOr2

[WeakenOr2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ FromContradiction ⊳ \underline{a} ⊳ ˙¬ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⊢ MP⊳\underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⊳\underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[WeakenOr2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⇒ \underline{b}]

[WeakenOr2 →^{tex} “WeakenOr2”]

[WeakenOr2 →^{pyk} “prop lemma weaken or second”]

Formula2Pair

[Formula2Pair →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⊢ PairDef ≫ ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\}; IffFirst ⊳ ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊳ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ≫ \underline{s} ∈ \{\underline{x}, \underline{y}\}⌉, p_{0}, c)]

[Formula2Pair →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⊢ \underline{s} ∈ \{\underline{x}, \underline{y}\}]

[Formula2Pair →^{tex} “Formula2Pair”] [Formula2Pair →^{pyk} “lemma formula2pair”]

Pair2Formula

[Pair2Formula →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊢ PairDef ≫ ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\}; IffSecond ⊳ ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ ˙¬ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊳ \underline{s} ∈ \{\underline{x}, \underline{y}\} ≫ ˙¬ \underline{s}= \underline{x} ⇒ \underline{s}=\underline{y}⌉, p_{0}, c)]

[Pair2Formula →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊢ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{y}]

[Pair2Formula →^{tex} “Pair2Formula”]

[Pair2Formula →^{pyk} “lemma pair2formula”]

Formula2Union

[Formula2Union →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ j_{Ex} ⊢ j_{Ex} ∈ \underline{x} ⊢ JoinConjuncts ⊳ \underline{s} ∈ j_{Ex} ⊳ j_{Ex} ∈ \underline{x} ≫ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x}; UnionDef ≫ ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x}; IffFirst ⊳ ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x} ⊳ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ≫ \underline{s} ∈ ∪\underline{x}⌉, p_{0}, c)]

[Formula2Union →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ j_{Ex} ⊢ j_{Ex} ∈ \underline{x} ⊢ \underline{s} ∈ ∪\underline{x}]

[Formula2Union →^{tex} “Formula2Union”]

[Formula2Union →^{pyk} “lemma formula2union”]

Union2Formula

[Union2Formula →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ ∪\underline{x} ⊢ UnionDef ≫ ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x}; IffSecond ⊳ ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x} ⊳ \underline{s} ∈ ∪\underline{x} ≫ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x}⌉, p_{0}, c)]

[Union2Formula →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ ∪\underline{x} ⊢ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x}]

[Union2Formula →^{tex} “Union2Formula”]

[Union2Formula →^{pyk} “lemma union2formula”]

Formula2Sep

[Formula2Sep →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{y}: \underline{p}^{Ph}∧⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ⊢⊢ \underline{y} ∈ \underline{x} ⊢ \underline{b} ⊢ JoinConjuncts⊳\underline{y} ∈ \underline{x} ⊳ \underline{b} ≫ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b}; SeparationDef ⊳ ⊳ \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ≫ ˙¬ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}; SecondConjunct ⊳ ˙¬ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ≫ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}; MP ⊳ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊳ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ≫ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}⌉, p_{0}, c)]

[Formula2Sep →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{y}: \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ⊢⊢ \underline{y} ∈ \underline{x} ⊢ \underline{b} ⊢ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}]

[Formula2Sep →^{tex} “Formula2Sep”]

[Formula2Sep →^{pyk} “lemma formula2separation”]

Sep2Formula

[Sep2Formula →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{y}: \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ⊢⊢ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ SeparationDef ⊳ ⊳ \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ≫ ˙¬ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}; FirstConjunct ⊳ ˙¬ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ≫ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b}; MP ⊳ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⊳ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ≫ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b}⌉, p_{0}, c)]

[Sep2Formula →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{y}: \underline{p}^{Ph} ∧ ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{p}⌉:=⌈\underline{y}⌉⟩_{Ph} ⊢⊢ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ ˙¬ \underline{y} ∈ \underline{x} ⇒ ˙¬ \underline{b}]

[Sep2Formula →^{tex} “Sep2Formula”]

[Sep2Formula →^{pyk} “lemma separation2formula”]

SubsetInPower

[SubsetInPower →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⊢ Gen ⊳ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}; PowerDef ≫ ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}); IffFirst ⊳ ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}) ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ≫ \underline{s} ∈ P(\underline{x})⌉, p_{0}, c)]

[SubsetInPower →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⊢ \underline{s} ∈ P(\underline{x})]

[SubsetInPower →^{tex} “SubsetInPower”] [SubsetInPower →^{pyk} “lemma subset in power set”]

HelperPowerIsSub

[HelperPowerIsSub →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊢ Repetition ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ≫ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊢ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y}⌉, p_{0}, c)]

[HelperPowerIsSub →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y}]

[HelperPowerIsSub →^{tex} “HelperPowerIsSub”]

[HelperPowerIsSub →^{pyk} “lemma power set is subset0”]

PowerIsSub

[PowerIsSub →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ P(\underline{x}) ⊢ PowerDef ≫ ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}); IffSecond ⊳ ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}) ⊳ \underline{s} ∈ P(\underline{x}) ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}; HelperPowerIsSub ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}; MP ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ≫ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}; Repetition ⊳ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ≫ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}⌉, p_{0}, c)]

[PowerIsSub →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ P(\underline{x}) ⊢ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}]

[PowerIsSub →^{tex} “PowerIsSub”]

[PowerIsSub →^{pyk} “lemma power set is subset”]

(Switch)HelperPowerIsSub

[(Switch)HelperPowerIsSub →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ HelperPowerIsSub ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y}⌉, p_{0}, c)]

[(Switch)HelperPowerIsSub →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y}]

[(Switch)HelperPowerIsSub →^{tex} “(Switch)HelperPowerIsSub”]

[(Switch)HelperPowerIsSub →^{pyk} “lemma power set is subset0-switch”]

(Switch)PowerIsSub

[(Switch)PowerIsSub →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{s} ∈ P(\underline{x}) ⊢ PowerIsSub ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ \underline{s} ∈ P(\underline{x}) ≫ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}⌉, p_{0}, c)]

[(Switch)PowerIsSub →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{s} ∈ P(\underline{x}) ⊢ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x}]

[(Switch)PowerIsSub →^{tex} “(Switch)PowerIsSub”]

[(Switch)PowerIsSub →^{pyk} “lemma power set is subset-switch”]

ToSetEquality

[ToSetEquality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊢ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊢ JoinConjuncts ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊳ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; Gen ⊳ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; Extensionality ≫ ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y}; SecondConjunct ⊳ ˙¬ \underline{x}= \underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y} ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}= \underline{y}; MP ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y} ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ \underline{x}=\underline{y}⌉, p_{0}, c)]

[ToSetEquality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊢ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊢ \underline{x}=\underline{y}]

[ToSetEquality →^{tex} “ToSetEquality”]

[ToSetEquality →^{pyk} “lemma set equality suff condition”]

HelperToSetEquality(t)

[HelperToSetEquality(t) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⊢ Repetition ⊳ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ≫ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⊢ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ≫ ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}⌉, p_{0}, c)]

[HelperToSetEquality(t) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}]

[HelperToSetEquality(t) →^{tex} “HelperToSetEquality(t)”]

[HelperToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)0”]

ToSetEquality(t)

[ToSetEquality(t) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⊢ \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ⊢ Gen ⊳ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ≫ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y}; HelperToSetEquality(t) ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ≫ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}; MP ⊳ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊳ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ≫ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}; Gen ⊳ \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ≫ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x}; HelperToSetEquality(t) ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ≫ ∀_{obj}\overline{t}: \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; MP⊳∀_{obj}\overline{t}: \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}⊳∀_{obj}\overline{t}: \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ≫ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; ToSetEquality ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⊳ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ \underline{x}=\underline{y}⌉, p_{0}, c)]

[ToSetEquality(t) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{t}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \overline{t} ∈ \underline{x} ⇒ \overline{t} ∈ \underline{y} ⊢ \overline{t} ∈ \underline{y} ⇒ \overline{t} ∈ \underline{x} ⊢ \underline{x}=\underline{y}]

[ToSetEquality(t) →^{tex} “ToSetEquality(t)”]

[ToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)”]

HelperFromSetEquality

[HelperFromSetEquality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊢ Repetition ⊳ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊢ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x}⌉, p_{0}, c)]

[HelperFromSetEquality →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x}]

[HelperFromSetEquality →^{tex} “HelperFromSetEquality”]

[HelperFromSetEquality →^{pyk} “lemma set equality skip quantifier”]

FromSetEquality

[FromSetEquality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}= \underline{y} ⊢ \underline{s} ∈ \underline{x} ⊢ Extensionality ≫ ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y}; IffSecond ⊳ ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y} ⊳ \underline{x}=\underline{y} ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; HelperFromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x}; MP ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x}; IffSecond ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ \underline{s} ∈ \underline{x} ⊳ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ \underline{y}⌉, p_{0}, c)]

[FromSetEquality →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ \underline{y}]

[FromSetEquality →^{tex} “FromSetEquality”]

[FromSetEquality →^{pyk} “lemma set equality nec condition”]

HelperReflexivity

[HelperReflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊢ Repetition ⊳ \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r}; ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊢ \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[HelperReflexivity →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}]

[HelperReflexivity →^{tex} “HelperReflexivity”]

[HelperReflexivity →^{pyk} “lemma reflexivity0”]

Reflexivity

[Reflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ HelperReflexivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}; MP2 ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r} ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[Reflexivity →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}]

[Reflexivity →^{tex} “Reflexivity”]

[Reflexivity →^{pyk} “lemma reflexivity”]

HelperSymmetry

[HelperSymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊢ Repetition ⊳ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}; ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊢ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[HelperSymmetry →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r}]

[HelperSymmetry →^{tex} “HelperSymmetry”]

[HelperSymmetry →^{pyk} “lemma symmetry0”]

Symmetry

[Symmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{t} ∈ \underline{bs} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⊢ HelperSymmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r}; MP4 ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r} ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ⊳ \underline{t} ∈ \underline{bs} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ≫ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[Symmetry →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{t} ∈ \underline{bs} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⊢ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{s}\}\} ∈ \underline{r}]

[Symmetry →^{tex} “Symmetry”]

[Symmetry →^{pyk} “lemma symmetry”]

HelperTransitivity

[HelperTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ Repetition ⊳ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{u}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{bs}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \underline{u} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[HelperTransitivity →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{u}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \underline{u} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}]

[HelperTransitivity →^{tex} “HelperTransitivity”]

[HelperTransitivity →^{pyk} “lemma transitivity0”]

Transitivity

[Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{u}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{t} ∈ \underline{bs} ⊢ \underline{u} ∈ \underline{bs} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⊢ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⊢ HelperTransitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \underline{u} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}; MP5 ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{t} ∈ \underline{bs} ⇒ \underline{u} ∈ \underline{bs} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⇒ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r} ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}⊳\underline{s} ∈ \underline{bs}⊳\underline{t} ∈ \underline{bs}⊳\underline{u} ∈ \underline{bs}⊳\{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ≫ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}; MP ⊳ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r} ⊳ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[Transitivity →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{t}: ∀\underline{u}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{t} ∈ \underline{bs} ⊢ \underline{u} ∈ \underline{bs} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{t}\}\} ∈ \underline{r} ⊢ \{\{\underline{t}, \underline{t}\}, \{\underline{t}, \underline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{u}\}\} ∈ \underline{r}]

[Transitivity →^{tex} “Transitivity”]

[Transitivity →^{pyk} “lemma transitivity”]

ERisReflexive

[ERisReflexive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ Repetition ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[ERisReflexive →^{stmt} ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r}]

[ERisReflexive →^{tex} “ERisReflexive”]

[ERisReflexive →^{pyk} “lemma er is reflexive”]

ERisSymmetric

[ERisSymmetric →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ Repetition ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[ERisSymmetric →^{stmt} ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}]

[ERisSymmetric →^{tex} “ERisSymmetric”]

[ERisSymmetric →^{pyk} “lemma er is symmetric”]

ERisTransitive

[ERisTransitive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ Repetition ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[ERisTransitive →^{stmt} ZFsub ⊢ ∀\underline{r}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}]

[ERisTransitive →^{tex} “ERisTransitive”]

[ERisTransitive →^{pyk} “lemma er is transitive”]

ØisSubset

[ØisSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ Ø ⊢ Ødef ≫ ˙¬ \underline{s} ∈ Ø; FromContradiction ⊳ \underline{s} ∈ Ø ⊳ ˙¬ \underline{s} ∈ Ø ≫ \underline{s} ∈ \underline{x}; ∀\underline{s}: ∀\underline{x}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ Ø ⊢ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ Ø ⇒ \underline{s} ∈ \underline{x}⌉, p_{0}, c)]

[ØisSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: \underline{s} ∈ Ø ⇒ \underline{s} ∈ \underline{x}]

[ØisSubset →^{tex} “\O{}isSubset”]

[ØisSubset →^{pyk} “lemma empty set is subset”]

HelperMemberNotØ

[HelperMemberNotØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⊢ \underline{x}=Ø ⊢ FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ \underline{x}=Ø ⊳ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ Ø; ∀\underline{s}: ∀\underline{x}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⊢ \underline{x}=Ø ⊢ \underline{s} ∈ Ø ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⇒ \underline{x}=Ø ⇒ \underline{s} ∈ Ø⌉, p_{0}, c)]

[HelperMemberNotØ →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⇒ \underline{x}=Ø ⇒ \underline{s} ∈ Ø]

[HelperMemberNotØ →^{tex} “HelperMemberNot\O{}”]

[HelperMemberNotØ →^{pyk} “lemma member not empty0”]

MemberNotØ

[MemberNotØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⊢ HelperMemberNotØ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ≫ \underline{s} ∈ \underline{x} ⇒ \underline{x}=Ø ⇒ \underline{s} ∈ Ø; MP ⊳ \underline{s} ∈ \underline{x} ⇒ \underline{x}= Ø ⇒ \underline{s} ∈ Ø ⊳ \underline{s} ∈ \underline{x} ≫ \underline{x}=Ø ⇒ \underline{s} ∈ Ø; Ødef ≫ ˙¬ \underline{s} ∈ Ø; MT ⊳ \underline{x}=Ø ⇒ \underline{s} ∈ Ø ⊳ ˙¬ \underline{s} ∈ Ø ≫ ˙¬ \underline{x}=Ø⌉, p_{0}, c)]

[MemberNotØ →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ \underline{s} ∈ \underline{x} ⊢ ˙¬ \underline{x}=Ø]

[MemberNotØ →^{tex} “MemberNot\O{}”]

[MemberNotØ →^{pyk} “lemma member not empty”]

HelperUniqueØ

[HelperUniqueØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ \underline{x} ⊢ FromContradiction ⊳ \underline{s} ∈ \underline{x} ⊳ ˙¬ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ Ø; ∀\underline{s}: ∀\underline{x}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ Ø ≫ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ Ø; ˙¬ \underline{s} ∈ \underline{x} ⊢ MP ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ Ø ⊳ ˙¬ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ Ø⌉, p_{0}, c)] [HelperUniqueØ →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ Ø]

[HelperUniqueØ →^{tex} “HelperUnique\O{}”]

[HelperUniqueØ →^{pyk} “lemma unique empty set0”]

UniqueØ

[UniqueØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \overline{s} ∈ \underline{x} ⊢ HelperUniqueØ ⊳ ˙¬ \overline{s} ∈ \underline{x} ≫ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ Ø; ØisSubset ≫ \overline{s} ∈ Ø ⇒ \overline{s} ∈ \underline{x}; ToSetEquality ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ Ø ⊳ \overline{s} ∈ Ø ⇒ \overline{s} ∈ \underline{x} ≫ \underline{x}=Ø⌉, p_{0}, c)]

[UniqueØ →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \overline{s} ∈ \underline{x} ⊢ \underline{x}=Ø]

[UniqueØ →^{tex} “Unique\O{}”]

[UniqueØ →^{pyk} “lemma unique empty set”]

=Reflexivity

[=Reflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: AutoImply ≫ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{s}; ToSetEquality ⊳ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{s} ⊳ \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{s} ≫ \underline{s}=\underline{s}⌉, p_{0}, c)]

[=Reflexivity →^{stmt} ZFsub ⊢ ∀\underline{s}: \underline{s}=\underline{s}]

[=Reflexivity →^{tex} “=\!{}Reflexivity”]

[=Reflexivity →^{pyk} “lemma =reflexivity”]

=Symmetry

[=Symmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ Extensionality ≫ ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y}; IffSecond ⊳ ˙¬ \underline{x}=\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}=\underline{y} ⊳ \underline{x}=\underline{y} ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; HelperFromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ≫ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; MP ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊳ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; FirstConjunct ⊳ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y}; SecondConjunct ⊳ ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ≫ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x}; ToSetEquality ⊳ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ≫ \underline{y}=\underline{x}⌉, p_{0}, c)] [=Symmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{y}=\underline{x}]

[=Symmetry →^{tex} “=\!{}Symmetry”]

[=Symmetry →^{pyk} “lemma =symmetry”]

Helper=Transitivity

[Helper=Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{y}=\underline{z} ⊢ \underline{s} ∈ \underline{x} ⊢ FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ⊳ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ \underline{y}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊳ \underline{y}=\underline{z} ⊳ \underline{s} ∈ \underline{y} ≫ \underline{s} ∈ \underline{z}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{y}=\underline{z} ⊢ \underline{s} ∈ \underline{x} ⊢ \underline{s} ∈ \underline{z} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{y}=\underline{z} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{z}⌉, p_{0}, c)]

[Helper=Transitivity →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{y}=\underline{z} ⇒ \underline{s} ∈ \underline{x} ⇒ \underline{s} ∈ \underline{z}]

[Helper=Transitivity →^{tex} “Helper\!{}=\!{}Transitivity”]

[Helper=Transitivity →^{pyk} “lemma =transitivity0”]

=Transitivity

[=Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{y}=\underline{z} ⊢ Helper=Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ≫ \underline{x}=\underline{y} ⇒ \underline{y}=\underline{z} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{z}; MP2 ⊳ \underline{x}=\underline{y} ⇒ \underline{y}=\underline{z} ⇒ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{z} ⊳ \underline{x}=\underline{y} ⊳ \underline{y}=\underline{z} ≫ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{z}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}= \underline{y} ≫ \underline{y}=\underline{x}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊳ \underline{y}=\underline{z} ≫ \underline{z}=\underline{y}; Helper= Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ≫ \underline{z}=\underline{y} ⇒ \underline{y}=\underline{x} ⇒ \overline{s} ∈ \underline{z} ⇒ \overline{s} ∈ \underline{x}; MP2 ⊳ \underline{z}=\underline{y} ⇒ \underline{y}=\underline{x} ⇒ \overline{s} ∈ \underline{z} ⇒ \overline{s} ∈ \underline{x} ⊳ \underline{z}=\underline{y} ⊳ \underline{y}=\underline{x} ≫ \overline{s} ∈ \underline{z} ⇒ \overline{s} ∈ \underline{x}; ToSetEquality ⊳ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{z} ⊳ \overline{s} ∈ \underline{z} ⇒ \overline{s} ∈ \underline{x} ≫ \underline{x}=\underline{z}⌉, p_{0}, c)]

[=Transitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{z}⌉ ⊢⊢ \underline{x}= \underline{y} ⊢ \underline{y}=\underline{z} ⊢ \underline{x}=\underline{z}]

[=Transitivity →^{tex} “\!{}=\!{}Transitivity”]

[=Transitivity →^{pyk} “lemma =transitivity”]

HelperTransferNotEq

[HelperTransferNotEq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{v} ⊢ \underline{y}=\underline{w} ⊢ \underline{v}=\underline{w} ⊢ =Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ \underline{x}=\underline{v} ⊳ \underline{v}=\underline{w} ≫ \underline{x}=\underline{w}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ \underline{y}=\underline{w} ≫ \underline{w}=\underline{y}; =Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{w} ⊳ \underline{w}=\underline{y} ≫ \underline{x}= \underline{y}; ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{v} ⊢ \underline{y}=\underline{w} ⊢ \underline{v}=\underline{w} ⊢ \underline{x}=\underline{y} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{v} ⇒ \underline{y}=\underline{w} ⇒ \underline{v}=\underline{w} ⇒ \underline{x}=\underline{y}⌉, p_{0}, c)]

[HelperTransferNotEq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{v} ⇒ \underline{y}=\underline{w} ⇒ \underline{v}=\underline{w} ⇒ \underline{x}=\underline{y}]

[HelperTransferNotEq →^{tex} “HelperTransferNotEq”]

[HelperTransferNotEq →^{pyk} “lemma transfer ˜ is0”]

TransferNotEq

[TransferNotEq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ˙¬ \underline{x}=\underline{y} ⊢ \underline{x}=\underline{v} ⊢ \underline{y}=\underline{w} ⊢ HelperTransferNotEq ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ≫ \underline{x}=\underline{v} ⇒ \underline{y}=\underline{w} ⇒ \underline{v}=\underline{w} ⇒ \underline{x}= \underline{y}; MP2 ⊳ \underline{x}=\underline{v} ⇒ \underline{y}=\underline{w} ⇒ \underline{v}=\underline{w} ⇒ \underline{x}=\underline{y} ⊳ \underline{x}=\underline{v} ⊳ \underline{y}=\underline{w} ≫ \underline{v}=\underline{w} ⇒ \underline{x}= \underline{y}; MT ⊳ \underline{v}=\underline{w} ⇒ \underline{x}=\underline{y} ⊳ ˙¬ \underline{x}=\underline{y} ≫ ˙¬ \underline{v}=\underline{w}⌉, p_{0}, c)]

[TransferNotEq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ˙¬ \underline{x}=\underline{y} ⊢ \underline{x}=\underline{v} ⊢ \underline{y}=\underline{w} ⊢ ˙¬ \underline{v}=\underline{w}]

[TransferNotEq →^{tex} “TransferNotEq”]

[TransferNotEq →^{pyk} “lemma transfer ˜ is”]

HelperPairSubset

[HelperPairSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{s}=\underline{x} ⊢ =Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{s}=\underline{x} ⊳ \underline{x}=\underline{y} ≫ \underline{s}=\underline{y}; WeakenOr2 ⊳ \underline{s}=\underline{y} ≫ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{w}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{s}=\underline{x} ⊢ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}⌉, p_{0}, c)]

[HelperPairSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}]

[HelperPairSubset →^{tex} “HelperPairSubset”]

[HelperPairSubset →^{pyk} “lemma pair subset0”]

Helper(2)PairSubset

[Helper(2)PairSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{v}=\underline{w} ⊢ \underline{s}=\underline{v} ⊢ =Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ \underline{s}=\underline{v} ⊳ \underline{v}=\underline{w} ≫ \underline{s}=\underline{w}; WeakenOr1 ⊳ \underline{s}= \underline{w} ≫ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; ∀\underline{s}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: Ded ⊳ ∀\underline{s}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{v}=\underline{w} ⊢ \underline{s}=\underline{v} ⊢ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{v}=\underline{w} ⇒ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}⌉, p_{0}, c)]

[Helper(2)PairSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{v}=\underline{w} ⇒ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}]

[Helper(2)PairSubset →^{tex} “Helper(2)PairSubset”]

[Helper(2)PairSubset →^{pyk} “lemma pair subset1”]

PairSubset

[PairSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ \underline{s} ∈ \{\underline{x}, \underline{v}\} ⊢ Pair2Formula ⊳ \underline{s} ∈ \{\underline{x}, \underline{v}\} ≫ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{v}; HelperPairSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ≫ \underline{x}=\underline{y} ⇒ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; MP ⊳ \underline{x}= \underline{y} ⇒ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ⊳ \underline{x}=\underline{y} ≫ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}= \underline{w}; Helper(2)PairSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ≫ \underline{v}=\underline{w} ⇒ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; MP ⊳ \underline{v}=\underline{w} ⇒ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ⊳ \underline{v}= \underline{w} ≫ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; FromDisjuncts ⊳ ˙¬ \underline{s}=\underline{x} ⇒ \underline{s}=\underline{v} ⊳ \underline{s}=\underline{x} ⇒ ˙¬ \underline{s}= \underline{y} ⇒ \underline{s}=\underline{w} ⊳ \underline{s}=\underline{v} ⇒ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ≫ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w}; Formula2Pair ⊳ ˙¬ \underline{s}=\underline{y} ⇒ \underline{s}=\underline{w} ≫ \underline{s} ∈ \{\underline{y}, \underline{w}\}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ \underline{s} ∈ \{\underline{x}, \underline{v}\} ⊢ \underline{s} ∈ \{\underline{y}, \underline{w}\} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{v}=\underline{w} ⇒ \underline{s} ∈ \{\underline{x}, \underline{v}\} ⇒ \underline{s} ∈ \{\underline{y}, \underline{w}\}⌉, p_{0}, c)]

[PairSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{v}=\underline{w} ⇒ \underline{s} ∈ \{\underline{x}, \underline{v}\} ⇒ \underline{s} ∈ \{\underline{y}, \underline{w}\}]

[PairSubset →^{tex} “PairSubset”]

[PairSubset →^{pyk} “lemma pair subset”]

SamePair

[SamePair →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{w}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ PairSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ≫ \underline{x}=\underline{y} ⇒ \underline{v}= \underline{w} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\}; MP2 ⊳ \underline{x}=\underline{y} ⇒ \underline{v}=\underline{w} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⊳ \underline{x}=\underline{y} ⊳ \underline{v}=\underline{w} ≫ \overline{t} ∈ \{\underline{x}, \underline{v}\} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ≫ \underline{y}=\underline{x}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ \underline{v}=\underline{w} ≫ \underline{w}= \underline{v}; PairSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ≫ \underline{y}=\underline{x} ⇒ \underline{w}= \underline{v} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\}; MP2 ⊳ \underline{y}=\underline{x} ⇒ \underline{w}=\underline{v} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\} ⊳ \underline{y}=\underline{x} ⊳ \underline{w}=\underline{v} ≫ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\}; ToSetEquality(t) ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{v}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{w}\}⌉ ⊳ \overline{t} ∈ \{\underline{x}, \underline{v}\} ⇒ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⊳ \overline{t} ∈ \{\underline{y}, \underline{w}\} ⇒ \overline{t} ∈ \{\underline{x}, \underline{v}\} ≫ \{\underline{x}, \underline{v}\}=\{\underline{y}, \underline{w}\}⌉, p_{0}, c)]

[SamePair →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{w}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ \{\underline{x}, \underline{v}\}=\{\underline{y}, \underline{w}\}]

[SamePair →^{tex} “SamePair”]

[SamePair →^{pyk} “lemma same pair”]

SameSingleton

[SameSingleton →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ SamePair ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊳ \underline{x}=\underline{y} ⊳ \underline{x}=\underline{y} ≫ \{\underline{x}, \underline{x}\}=\{\underline{y}, \underline{y}\}; Repetition ⊳ \{\underline{x}, \underline{x}\}=\{\underline{y}, \underline{y}\} ≫ \{\underline{x}, \underline{x}\}=\{\underline{y}, \underline{y}\}⌉, p_{0}, c)]

[SameSingleton →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \{\underline{x}, \underline{x}\}=\{\underline{y}, \underline{y}\}]

[SameSingleton →^{tex} “SameSingleton”]

[SameSingleton →^{pyk} “lemma same singleton”]

UnionSubset

[UnionSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{s} ∈ ∪\underline{x} ⊢ Union2Formula ⊳ \underline{s} ∈ ∪\underline{x} ≫ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ≫ \underline{s} ∈ j_{Ex}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ≫ j_{Ex} ∈ \underline{x}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}= \underline{y} ⊳ j_{Ex} ∈ \underline{x} ≫ j_{Ex} ∈ \underline{y}; Formula2Union ⊳ \underline{s} ∈ j_{Ex} ⊳ j_{Ex} ∈ \underline{y} ≫ \underline{s} ∈ ∪\underline{y}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{s} ∈ ∪\underline{x} ⊢ \underline{s} ∈ ∪\underline{y} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{s} ∈ ∪\underline{x} ⇒ \underline{s} ∈ ∪\underline{y}⌉, p_{0}, c)]

[UnionSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{s} ∈ ∪\underline{x} ⇒ \underline{s} ∈ ∪\underline{y}]

[UnionSubset →^{tex} “UnionSubset”]

[UnionSubset →^{pyk} “lemma union subset”]

SameUnion

[SameUnion →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ UnionSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ≫ \underline{x}=\underline{y} ⇒ \overline{s} ∈ ∪\underline{x} ⇒ \overline{s} ∈ ∪\underline{y}; MP ⊳ \underline{x}= \underline{y} ⇒ \overline{s} ∈ ∪\underline{x} ⇒ \overline{s} ∈ ∪\underline{y} ⊳ \underline{x}=\underline{y} ≫ \overline{s} ∈ ∪\underline{x} ⇒ \overline{s} ∈ ∪\underline{y}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ≫ \underline{y}=\underline{x}; UnionSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ≫ \underline{y}=\underline{x} ⇒ \overline{s} ∈ ∪\underline{y} ⇒ \overline{s} ∈ ∪\underline{x}; MP ⊳ \underline{y}=\underline{x} ⇒ \overline{s} ∈ ∪\underline{y} ⇒ \overline{s} ∈ ∪\underline{x} ⊳ \underline{y}=\underline{x} ≫ \overline{s} ∈ ∪\underline{y} ⇒ \overline{s} ∈ ∪\underline{x}; ToSetEquality ⊳ \overline{s} ∈ ∪\underline{x} ⇒ \overline{s} ∈ ∪\underline{y} ⊳ \overline{s} ∈ ∪\underline{y} ⇒ \overline{s} ∈ ∪\underline{x} ≫ ∪\underline{x}=∪\underline{y}⌉, p_{0}, c)]

[SameUnion →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ ∪\underline{x}=∪\underline{y}]

[SameUnion →^{tex} “SameUnion”]

[SameUnion →^{pyk} “lemma same union”]

SeparationSubset

[SeparationSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ≫ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{a}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{a} ≫ \underline{s} ∈ \underline{x}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ⊳ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ \underline{y}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{a} ≫ \underline{a}; IffSecond ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊳ \underline{a} ≫ \underline{b}; Formula2Sep ⊳ \underline{s} ∈ \underline{y} ⊳ \underline{b} ≫ \underline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}; ∀\underline{a}: ∀\underline{b}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ \underline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ \underline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}⌉, p_{0}, c)]

[SeparationSubset →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}= \underline{y} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ \underline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}]

[SeparationSubset →^{tex} “SeparationSubset”]

[SeparationSubset →^{pyk} “lemma separation subset”]

SameSeparation

[SameSeparation →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ SeparationSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ≫ \underline{x}= \underline{y} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}; MP2 ⊳ \underline{x}=\underline{y} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ⊳ \underline{x}=\underline{y} ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}; =Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ≫ \underline{y}=\underline{x}; IffCommutativity ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; SeparationSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ≫ \underline{y}=\underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\}; MP2⊳\underline{y}=\underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊳ \underline{y}=\underline{x} ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ≫ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\}; ToSetEquality ⊳ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\}⊳\overline{s} ∈ \{ph ∈ \underline{y} | \underline{b}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{x} | \underline{a}\} ≫ \{ph ∈ \underline{x} | \underline{a}\}=\{ph ∈ \underline{y} | \underline{b}\}⌉, p_{0}, c)]

[SameSeparation →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \{ph ∈ \underline{x} | \underline{a}\}=\{ph ∈ \underline{y} | \underline{b}\}]

[SameSeparation →^{tex} “SameSeparation”]

[SameSeparation →^{pyk} “lemma same separation”]

SameBinaryUnion

[SameBinaryUnion →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ SameSingleton ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊳ \underline{x}=\underline{y} ≫ \{\underline{x}, \underline{x}\}=\{\underline{y}, \underline{y}\}; SameSingleton ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊳ \underline{v}=\underline{w} ≫ \{\underline{v}, \underline{v}\}= \{\underline{w}, \underline{w}\}; SamePair ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊳ \{\underline{x}, \underline{x}\}= \{\underline{y}, \underline{y}\} ⊳ \{\underline{v}, \underline{v}\}=\{\underline{w}, \underline{w}\} ≫ \{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}=\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}; SameUnion ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}= \{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\} ≫ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}= ∪\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}; Repetition ⊳ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}=∪\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\} ≫ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}=∪\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉, p_{0}, c)]

[SameBinaryUnion →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{x}, \underline{x}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{y}, \underline{y}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{v}, \underline{v}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\underline{w}, \underline{w}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{v}, \underline{v}\}\}=∪\{\{\underline{y}, \underline{y}\}, \{\underline{w}, \underline{w}\}\}]

[SameBinaryUnion →^{tex} “SameBinaryUnion”]

[SameBinaryUnion →^{pyk} “lemma same binary union”]

IntersectionSubset

[IntersectionSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ⊢ FirstConjunct ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ≫ \underline{s} ∈ \underline{x}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊳ \underline{x}=\underline{y} ⊳ \underline{s} ∈ \underline{x} ≫ \underline{s} ∈ \underline{y}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ≫ \underline{s} ∈ \underline{v}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊳ \underline{v}=\underline{w} ⊳ \underline{s} ∈ \underline{v} ≫ \underline{s} ∈ \underline{w}; JoinConjuncts ⊳ \underline{s} ∈ \underline{y} ⊳ \underline{s} ∈ \underline{w} ≫ ˙¬ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{w}; ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: Ded ⊳ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⊢ \underline{v}=\underline{w} ⊢ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ⊢ ˙¬ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{w} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{v}=\underline{w} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{w}⌉, p_{0}, c)]

[IntersectionSubset →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{v}: ∀\underline{w}: ⌈\overline{s}⌉#^{0}⌈\underline{x}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{y}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{v}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{w}⌉ ⊢⊢ \underline{x}=\underline{y} ⇒ \underline{v}=\underline{w} ⇒ ˙¬ \underline{s} ∈ \underline{x} ⇒ ˙¬ \underline{s} ∈ \underline{v} ⇒ ˙¬ \underline{s} ∈ \underline{y} ⇒ ˙¬ \underline{s} ∈ \underline{w}]

[IntersectionSubset →^{tex} “IntersectionSubset”]

[IntersectionSubset →^{pyk} “lemma intersection subset”]

SameIntersection

[SameIntersection →^{proof} λc.λx.P(⌈ZFsub ⊢ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊢ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⊢ SameBinaryUnion ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊳ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\}=∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\}; IntersectionSubset ≫ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ \overline{y}= \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}; MP2 ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊳ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}; =Symmetry ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ≫ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x}; = Symmetry ⊳ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y}; IntersectionSubset ≫ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ⇒ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}= \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}; MP2 ⊳ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ⇒ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}= \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⊳ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ⊳ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}= \overline{y} ≫ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}; JoinConjuncts⊳ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⊳ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ≫ ˙¬ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}; SameSeparation ⊳ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\}=∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} ⊳ ˙¬ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y} ≫ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}; Repetition ⊳ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\} ≫ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}⌉, p_{0}, c)]

[SameIntersection →^{stmt} ZFsub ⊢ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊢ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ⊢ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}]

[SameIntersection →^{tex} “SameIntersection”]

[SameIntersection →^{pyk} “lemma same intersection”]

AutoMember

[AutoMember →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ ERisReflexive ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r}; Reflexivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r}; Formula2Sep ⊳ \underline{s} ∈ \underline{bs} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{s}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}⌉, p_{0}, c)]

[AutoMember →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}]

[AutoMember →^{tex} “AutoMember”]

[AutoMember →^{pyk} “lemma auto member”]

HelperEqSysNotØ

[HelperEqSysNotØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ ˙¬ \underline{s} ∈ P(\underline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ P(\underline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s} ≫ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s}; FirstConjunct ⊳ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s} ≫ a_{Ex} ∈ \underline{bs}; SecondConjunct ⊳ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}= \underline{s} ≫ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s}; AutoMember ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ a_{Ex} ∈ \underline{bs} ≫ a_{Ex} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}; FromSetEquality ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\underline{s} ⊳ a_{Ex} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\} ≫ a_{Ex} ∈ \underline{s}; MemberNotØ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ a_{Ex} ∈ \underline{s} ≫ ˙¬ \underline{s}=Ø; ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ ˙¬ \underline{s}=Ø ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}= b_{Ph}\} ⇒ ˙¬ \underline{s}=Ø⌉, p_{0}, c)]

[HelperEqSysNotØ →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{s}=Ø]

[HelperEqSysNotØ →^{tex} “HelperEqSysNot\O{}”]

[HelperEqSysNotØ →^{pyk} “lemma eq-system not empty0”]

EqSysNotØ

[EqSysNotØ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ HelperEqSysNotØ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{s}= Ø; MP ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}= b_{Ph}\} ⇒ ˙¬ \underline{s}=Ø ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{s}=Ø⌉, p_{0}, c)]

[EqSysNotØ →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{s}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{s}=Ø]

[EqSysNotØ →^{tex} “EqSysNot\O{}”]

[EqSysNotØ →^{pyk} “lemma eq-system not empty”]

HelperEqSubset

[HelperEqSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⊢ Repetition⊳\underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}; Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r}; ERisTransitive ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \underline{bs}; Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r}; Formula2Sep ⊳ \underline{s} ∈ \underline{bs} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: Ded⊳∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⊢ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⌉, p_{0}, c)]

[HelperEqSubset →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}]

[HelperEqSubset →^{tex} “HelperEqSubset”]

[HelperEqSubset →^{pyk} “lemma eq subset0”]

EqSubset

[EqSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ HelperEqSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; MP4 ⊳ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⊳\underline{x} ∈ \underline{bs}⊳\underline{y} ∈ \underline{bs}⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⌉, p_{0}, c)]

[EqSubset →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}]

[EqSubset →^{tex} “EqSubset”]

[EqSubset →^{pyk} “lemma eq subset”]

HelperEqNecessary

[HelperEqNecessary →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ EqSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; ERisSymmetric ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}; Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ \{\{\underline{y}, \underline{y}\}, \{\underline{y}, \underline{x}\}\} ∈ \underline{r}; EqSubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊳ \underline{y} ∈ \underline{bs} ⊳ \underline{x} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \{\{\underline{y}, \underline{y}\}, \{\underline{y}, \underline{x}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}; ToSetEquality ⊳ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⊳ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ≫ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⌉, p_{0}, c)]

[HelperEqNecessary →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}]

[HelperEqNecessary →^{tex} “HelperEqNecessary”]

[HelperEqNecessary →^{pyk} “lemma equivalence nec condition0”]

EqNecessary

[EqNecessary →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ HelperEqNecessary ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; MP3 ⊳ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⌉, p_{0}, c)]

[EqNecessary →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}]

[EqNecessary →^{tex} “EqNecessary”]

[EqNecessary →^{pyk} “lemma equivalence nec condition”]

HelperNoneEqNecessary

[HelperNoneEqNecessary →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⊢ ERisSymmetric⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r}; ERisTransitive ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r}; Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ≫ ˙¬ \underline{s} ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} ⇒ ˙¬ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} ⇒ ˙¬ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}; Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r} ≫ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \underline{bs}; Symmetry ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ⊳ \underline{x} ∈ \underline{bs} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{s}\}\} ∈ \underline{r}; Transitivity ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{s} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{s}\}\} ∈ \underline{r} ⊳ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{y}\}\} ∈ \underline{r} ≫ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}; ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⊢ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}⌉, p_{0}, c)]

[HelperNoneEqNecessary →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}]

[HelperNoneEqNecessary →^{tex} “HelperNoneEqNecessary”]

[HelperNoneEqNecessary →^{pyk} “lemma none-equivalence nec condition0”]

Helper(2)NoneEqNecessary

[Helper(2)NoneEqNecessary →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ HelperNoneEqNecessary ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}; MP3 ⊳ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}; MT ⊳ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ⇒ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊳ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ ˙¬ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}; UniqueØ ⊳ ˙¬ \overline{s} ∈ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\} ≫ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø; ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⊢ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø⌉, p_{0}, c)]

[Helper(2)NoneEqNecessary →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø]

[Helper(2)NoneEqNecessary →^{tex} “Helper(2)NoneEqNecessary”]

[Helper(2)NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition1”]

NoneEqNecessary

[NoneEqNecessary →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ Helper(2)NoneEqNecessary ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø; MP3 ⊳ \underline{x} ∈ \underline{bs} ⇒ \underline{y} ∈ \underline{bs} ⇒ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø⌉, p_{0}, c)]

[NoneEqNecessary →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø]

[NoneEqNecessary →^{tex} “NoneEqNecessary”]

[NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition”]

EqClassIsSubset

[EqClassIsSubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{bs}: \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⊢ Sep2Formula ⊳ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{\{\underline{s}, \underline{s}\}, \{\underline{s}, \underline{x}\}\} ∈ \underline{r} ≫ \underline{s} ∈ \underline{bs}; ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{bs}: \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⊢ \underline{s} ∈ \underline{bs} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \underline{bs}⌉, p_{0}, c)]

[EqClassIsSubset →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{x}: ∀\underline{bs}: \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ \underline{s} ∈ \underline{bs}]

[EqClassIsSubset →^{tex} “EqClassIsSubset”]

[EqClassIsSubset →^{pyk} “lemma equivalence class is subset”]

EqClassesAreDisjoint

[EqClassesAreDisjoint →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⊢ EqNecessary ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊳ \underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}; MT ⊳ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}⊳ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ≫ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r}; NoneEqNecessary ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉⊳\underline{x} ∈ \underline{bs} ⊳ \underline{y} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø; MP ⊳ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ⇒ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø ⊳ ˙¬ \{\{\underline{x}, \underline{x}\}, \{\underline{x}, \underline{y}\}\} ∈ \underline{r} ≫ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø⌉, p_{0}, c)]

[EqClassesAreDisjoint →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ \underline{x} ∈ \underline{bs} ⊢ \underline{y} ∈ \underline{bs} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\} ⊢ \{ph ∈ ∪\{\{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\}\}, \{\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}, \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{x}\}\} ∈ \underline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{y}\}\} ∈ \underline{r}\}\}=Ø]

[EqClassesAreDisjoint →^{tex} “EqClassesAreDisjoint”]

[EqClassesAreDisjoint →^{pyk} “lemma equivalence classes are disjoint”]

AllDisjoint

[AllDisjoint →^{proof} λc.λx.P(⌈ZFsub ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⊢ \overline{x} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⊢ \overline{y} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⊢ ˙¬ \overline{x}=\overline{y} ⊢ Sep2Formula ⊳ \overline{x} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ≫ ˙¬ \overline{x} ∈ P(\overline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x}; SecondConjunct ⊳ ˙¬ \overline{x} ∈ P(\overline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ≫ ˙¬ a_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x}; FirstConjunct ⊳ ˙¬ a_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ≫ a_{Ex} ∈ \overline{bs}; SecondConjunct ⊳ ˙¬ a_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}= \overline{x} ≫ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x}; =Symmetry ⊳ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\overline{x} ≫ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}; Sep2Formula ⊳ \overline{y} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ≫ ˙¬ \overline{y} ∈ P(\overline{bs}) ⇒ ˙¬ ˙¬ b_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y}; SecondConjunct ⊳ ˙¬ \overline{y} ∈ P(\overline{bs}) ⇒ ˙¬ ˙¬ b_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y} ≫ ˙¬ b_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y}; FirstConjunct ⊳ ˙¬ b_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y} ≫ b_{Ex} ∈ \overline{bs}; SecondConjunct ⊳ ˙¬ b_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}= \overline{y} ≫ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y}; =Symmetry ⊳ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}=\overline{y} ≫ \overline{y}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}; TransferNotEq ⊳ ˙¬ \overline{x}=\overline{y} ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊳ \overline{y}= \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}; EqClassesAreDisjoint ⊳ a_{Ex} ∈ \overline{bs} ⊳ b_{Ex} ∈ \overline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⊳ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ \{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}=Ø; SameIntersection ⊳ \overline{x}=\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⊳ \overline{y}= \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\} ≫ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}; =Transitivity ⊳ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=\{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\} ⊳ \{ph ∈ ∪\{\{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\}\}, \{\{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}, \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}\} | ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \overline{r}\} ⇒ ˙¬ c_{Ph} ∈ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, b_{Ex}\}\} ∈ \overline{r}\}\}=Ø ≫ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=Ø⌉, p_{0}, c)]

[AllDisjoint →^{stmt} ZFsub ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⊢ \overline{x} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⊢ \overline{y} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⊢ ˙¬ \overline{x}=\overline{y} ⊢ \{ph ∈ ∪\{\{\overline{x}, \overline{x}\}, \{\overline{y}, \overline{y}\}\} | ˙¬ c_{Ph} ∈ \overline{x} ⇒ ˙¬ c_{Ph} ∈ \overline{y}\}=Ø]

[AllDisjoint →^{tex} “AllDisjoint”]

[AllDisjoint →^{pyk} “lemma all disjoint”]

AllDisjointImply

[AllDisjointImply →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ ˙¬ \underline{x}= \underline{y} ⊢ CheatAllDisjoint⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊳ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊳ ˙¬ \underline{x}=\underline{y} ≫ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø; ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ ˙¬ \underline{x}= \underline{y} ⊢ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{x}=\underline{y} ⇒ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø⌉, p_{0}, c)]

[AllDisjointImply →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{x}: ∀\underline{y}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{x} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \underline{y} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ ˙¬ \underline{x}=\underline{y} ⇒ \{ph ∈ ∪\{\{\underline{x}, \underline{x}\}, \{\underline{y}, \underline{y}\}\} | ˙¬ c_{Ph} ∈ \underline{x} ⇒ ˙¬ c_{Ph} ∈ \underline{y}\}=Ø]

[AllDisjointImply →^{tex} “AllDisjointImply”]

[AllDisjointImply →^{pyk} “lemma all disjoint-imply”]

BSsubset

[BSsubset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ AutoMember ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊳ \underline{s} ∈ \underline{bs} ≫ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}; EqClassIsSubset ≫ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \underline{bs}; SubsetInPower ⊳ \overline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ⇒ \overline{s} ∈ \underline{bs} ≫ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ∈ P(\underline{bs}); =Reflexivity ≫ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}; JoinConjuncts ⊳ \underline{s} ∈ \underline{bs} ⊳ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ≫ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}; ExistIntro @ a_{Ex} @ \underline{s} ⊳ ˙¬ \underline{s} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}= \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ≫ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\}; Formula2Sep ⊳ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ∈ P(\underline{bs}) ⊳ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ≫ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; ExistIntro @ j_{Ex} @\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ⊳ \underline{s} ∈ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ≫ \underline{s} ∈ j_{Ex}; ExistIntro @ j_{Ex} @\{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ⊳ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, \underline{s}\}\} ∈ \underline{r}\} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; Formula2Union ⊳ \underline{s} ∈ j_{Ex} ⊳ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ \underline{bs} ⊢ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}⌉, p_{0}, c)]

[BSsubset →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ \underline{bs} ⇒ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}]

[BSsubset →^{tex} “BSsubset”]

[BSsubset →^{pyk} “lemma bs subset union(bs/r)”]

Union(BS/R)subset

[Union(BS/R)subset →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ Union2Formula ⊳ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}= b_{Ph}\}; FirstConjunct ⊳ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ \underline{s} ∈ j_{Ex}; SecondConjunct ⊳ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; Sep2Formula ⊳ j_{Ex} ∈ \{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ ˙¬ j_{Ex} ∈ P(\underline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}= j_{Ex}; FirstConjunct ⊳ ˙¬ j_{Ex} ∈ P(\underline{bs}) ⇒ ˙¬ ˙¬ a_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, a_{Ex}\}\} ∈ \underline{r}\}=j_{Ex} ≫ j_{Ex} ∈ P(\underline{bs}); (Switch)PowerIsSub ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ j_{Ex} ∈ P(\underline{bs}) ≫ \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs}; Gen ⊳ \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs} ≫ ∀_{obj}\overline{s}: \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs}; (Switch)HelperPowerIsSub ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ≫ ∀_{obj}\overline{s}: \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs} ⇒ \underline{s} ∈ j_{Ex} ⇒ \underline{s} ∈ \underline{bs}; MP2 ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs} ⇒ \underline{s} ∈ j_{Ex} ⇒ \underline{s} ∈ \underline{bs} ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ j_{Ex} ⇒ \overline{s} ∈ \underline{bs} ⊳ \underline{s} ∈ j_{Ex} ≫ \underline{s} ∈ \underline{bs}; ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: Ded ⊳ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊢ \underline{s} ∈ \underline{bs} ≫ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \underline{s} ∈ \underline{bs}⌉, p_{0}, c)]

[Union(BS/R)subset →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{s}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \underline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \underline{s} ∈ \underline{bs}]

[Union(BS/R)subset →^{tex} “Union(BS/R)subset”]

[Union(BS/R)subset →^{pyk} “lemma union(bs/r) subset bs”]

UnionIdentity

[UnionIdentity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{r}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ BSsubset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ \underline{bs} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; MP ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ \underline{bs} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \overline{s} ∈ \underline{bs} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}; Union(BS/R)subset ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊳ ⊳ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}= b_{Ph}\} ⇒ \overline{s} ∈ \underline{bs}; MP ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \overline{s} ∈ \underline{bs} ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ≫ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \overline{s} ∈ \underline{bs}; ToSetEquality ⊳ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ⇒ \overline{s} ∈ \underline{bs} ⊳ \overline{s} ∈ \underline{bs} ⇒ \overline{s} ∈ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\} ≫ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}= \underline{bs}⌉, p_{0}, c)]

[UnionIdentity →^{stmt} ZFsub ⊢ ∀\underline{r}: ∀\underline{bs}: ⌈\overline{s}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{s}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{t}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{r}⌉ ⊢⊢ ⌈\overline{u}⌉#^{0}⌈\underline{bs}⌉ ⊢⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \underline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \underline{bs} ⇒ \overline{t} ∈ \underline{bs} ⇒ \overline{u} ∈ \underline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \underline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \underline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \underline{r} ⊢ ∪\{ph ∈ P(\underline{bs}) | ˙¬ t_{Ex} ∈ \underline{bs} ⇒ ˙¬ \{ph ∈ \underline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \underline{r}\}=b_{Ph}\}=\underline{bs}]

[UnionIdentity →^{tex} “UnionIdentity”]

[UnionIdentity →^{pyk} “lemma union(bs/r) is bs”]

EqSysIsPartition

[EqSysIsPartition →^{proof} λc.λx.P(⌈ZFsub ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⊢ EqSysNotØ ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ≫ \overline{x} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{x}= Ø; Ded ⊳ \overline{x} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{x}=Ø ≫ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø; Gen ⊳ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ≫ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø; AllDisjointImply ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⇒ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø; MP ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⇒ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ≫ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø; Gen ⊳ \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ≫ ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø; Gen ⊳ ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}= b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ≫ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø; UnionIdentity ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ≫ ∪\{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs}; JoinConjuncts ⊳ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}= Ø ⊳ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ≫ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}= Ø; JoinConjuncts ⊳ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⊳ ∪\{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs} ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⇒ ˙¬ ∪ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs}; Repetition ⊳ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⇒ ˙¬ ∪ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs} ≫ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⇒ ˙¬ ∪ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs}⌉, p_{0}, c)]

[EqSysIsPartition →^{stmt} ZFsub ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{s}\}\} ∈ \overline{r} ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: ∀_{obj}\overline{u}: \overline{s} ∈ \overline{bs} ⇒ \overline{t} ∈ \overline{bs} ⇒ \overline{u} ∈ \overline{bs} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{t}\}\} ∈ \overline{r} ⇒ \{\{\overline{t}, \overline{t}\}, \{\overline{t}, \overline{u}\}\} ∈ \overline{r} ⇒ \{\{\overline{s}, \overline{s}\}, \{\overline{s}, \overline{u}\}\} ∈ \overline{r} ⊢ ˙¬ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=Ø ⇒ ˙¬ ∀_{obj}\overline{s}: ∀_{obj}\overline{t}: \overline{s} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ \overline{t} ∈ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\} ⇒ ˙¬ \overline{s}=\overline{t} ⇒ \{ph ∈ ∪\{\{\overline{s}, \overline{s}\}, \{\overline{t}, \overline{t}\}\} | ˙¬ c_{Ph} ∈ \overline{s} ⇒ ˙¬ c_{Ph} ∈ \overline{t}\}=Ø ⇒ ˙¬ ∪ \{ph ∈ P(\overline{bs}) | ˙¬ t_{Ex} ∈ \overline{bs} ⇒ ˙¬ \{ph ∈ \overline{bs} | \{\{a_{Ph}, a_{Ph}\}, \{a_{Ph}, t_{Ex}\}\} ∈ \overline{r}\}=b_{Ph}\}=\overline{bs}]

[EqSysIsPartition →^{tex} “EqSysIsPartition”]

[EqSysIsPartition →^{pyk} “theorem eq-system is partition”]

∗/∗

[bs/r →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[bs/r ¨= {ph ∈ P(bs) | Ex_{20} ∈ bs ˙∧ [Ex_{20} ∈ bs]_{r} = ph_{2}}]⌉)]

[x/y →^{tex} “#1.

/ #2.”]

[∗/∗ →^{pyk} “eq-system of " modulo "”]

∗ ∩ ∗

[x ∩ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∩ y ¨= {ph ∈ x ∪ y | ph_{3} ∈ x ˙∧ ph_{3} ∈ y}]⌉)]

[x ∩ y →^{tex} “#1.

\cap #2.”]

[∗ ∩ ∗ →^{pyk} “intersection " comma " end intersection”]

∪∗

[∪x →^{tex} “\cup #1.”]

[∪∗ →^{pyk} “union " end union”]

∗ ∪ ∗

[x ∪ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∪ y ¨= ∪{{x}, {y}}]⌉)]

[x ∪ y →^{tex} “#1.

\mathrel{\cup} #2.”] [∗ ∪ ∗ →^{pyk} “binary-union " comma " end union”]

P(∗)

[P(x) →^{tex} “P(#1.

)”] [P(∗) →^{pyk} “power " end power”]

{∗}

[{x} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[{x} ¨= {x, x}]⌉)]

[{x} →^{tex} “\{#1.

\}”] [{∗} →^{pyk} “zermelo singleton " end singleton”]

{∗, ∗}

[{x, y} →^{tex} “\{#1.

, #2. \}”] [{∗, ∗} →^{pyk} “zermelo pair " comma " end pair”]

⟨∗, ∗⟩

[⟨x, y⟩ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨x, y⟩ ¨= {{x}, {x, y}}]⌉)]

[⟨x, y⟩ →^{tex} “\langle #1. , #2. \rangle”]

[⟨∗, ∗⟩ →^{pyk} “zermelo ordered pair " comma " end pair”]

∗ ∈ ∗

[x ∈ y →^{tex} “#1.

\mathrel{\in} #2.”]

[∗ ∈ ∗ →^{pyk} “" zermelo in "”]

∗(∗, ∗)

[r(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[r(x, y) ¨= ⟨x, y⟩ ∈ r]⌉)]

[z(x, y) →^{tex} “#3.

(#1. , #2. )”]

[∗(∗, ∗) →^{pyk} “" is related to " under "”]

ReflRel(∗, ∗)

[ReflRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ReflRel(r, x) ¨= ∀s: (s ∈ x ⇒ r(s, s))]⌉)]

[ReflRel(r, x) →^{tex} “ReflRel(#1. , #2. )”]

[ReflRel(∗, ∗) →^{pyk} “" is reflexive relation in "”]

SymRel(∗, ∗)

[SymRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SymRel(r, x) ¨= ∀s, t: (s ∈ x ⇒ t ∈ x ⇒ r(s, t) ⇒ r(t, s))]⌉)] [SymRel(r, x) →^{tex} “SymRel(#1. , #2. )”]

[SymRel(∗, ∗) →^{pyk} “" is symmetric relation in "”]

TransRel(∗, ∗)

[TransRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TransRel(r, x) ¨= ∀s, t, u: (s ∈ x ⇒ t ∈ x ⇒ u ∈ x ⇒ r(s, t) ⇒ r(t, u) ⇒ r(s, u))]⌉)]

[TransRel(r, x) →^{tex} “TransRel(#1. , #2. )”]

[TransRel(∗, ∗) →^{pyk} “" is transitive relation in "”]

EqRel(∗, ∗)

[EqRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[EqRel(r, x) ¨= ReflRel(r, x) ˙∧ SymRel(r, x) ˙∧ TransRel(r, x)]⌉)]

[EqRel(r, x) →^{tex} “EqRel(#1. , #2. )”]

[EqRel(∗, ∗) →^{pyk} “" is equivalence relation in "”]

[∗ ∈ ∗]_{∗}

[[x ∈ bs]_{r} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[[x ∈ bs]_{r} ¨= {ph ∈ bs | r(ph_{1}, x)}]⌉)]

[[x ∈ bs]_{r} →^{tex} “[#1.

\mathrel{\in} #2. ] {#3. }”]

[[∗ ∈ ∗]_{∗} →^{pyk} “equivalence class of " in " modulo "”]

Partition(∗, ∗)

[Partition(p, bs) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Partition(p, bs) ¨= (∀s: (s ∈ p ⇒ s ≠ Ø)) ˙∧ (∀s, t: (s ∈ p ⇒ t ∈ p ⇒ s ≠ t ⇒ s ∩ t =Ø)) ˙∧ ∪ p=bs]⌉)]

[Partition(x, y) →^{tex} “Partition(#1. , #2. )”]

[Partition(∗, ∗) →^{pyk} “" is partition of "”]

∗=∗

[x=y →^{tex} “#1.

\!\mathrel{=}\! #2.”]

[∗=∗ →^{pyk} “" zermelo is "”]

∗ ⊆ ∗

[x ⊆ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ⊆ y ¨= (s ∈ x ⇒ s ∈ y)]⌉)]

[x ⊆ y →^{tex} “#1.

\mathrel{\subseteq} #2.”]

[∗ ⊆ ∗ →^{pyk} “" is subset of "”]

˙¬ ∗

[ ˙¬ x →^{tex} “\dot{\neg}\, #1.”]

[ ˙¬ ∗ →^{pyk} “not0 "”]

∗ ∉ ∗

[x ∉ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∉ y ¨= ˙¬ x ∈ y]⌉)]

[x ∉ y →^{tex} “#1.

\mathrel{\notin} #2.”]

[∗ ∉ ∗ →^{pyk} “" zermelo ˜ in "”]

∗ ≠ ∗

[x ≠ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ≠ y ¨= ˙¬ x=y]⌉)]

[x ≠ y →^{tex} “#1.

\mathrel{\neq} #2.”]

[∗ ≠ ∗ →^{pyk} “" zermelo ˜ is "”]

∗ ˙∧ ∗

[x ˙∧ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∧ y ¨= ˙¬ (x ⇒ ˙¬ y)]⌉)]

[x ˙∧ y →^{tex} “#1.

\mathrel{\dot{\wedge}} #2.”]

[∗ ˙∧ ∗ →^{pyk} “" and0 "”]

∗ ˙∨ ∗

[x ˙∨ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∨ y ¨= ˙¬ x ⇒ y]⌉)]

[x ˙∨ y →^{tex} “#1.

\mathrel{\dot{\vee}} #2.”]

[∗ ˙∨ ∗ →^{pyk} “" or0 "”]

∗ ˙⇔ ∗

[x ˙⇔ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙⇔ y ¨= (x ⇒ y) ˙∧ (y ⇒ x)]⌉)]

[x ˙⇔ y →^{tex} “#1.

\mathrel{\dot{\Leftrightarrow}} #2.”]

[∗ ˙⇔ ∗ →^{pyk} “" iff "”]

{ph ∈ ∗ | ∗}

[{ph ∈ x | a} →^{tex} “ \{ ph \mathrel{\in} #1. \mid #2. \}”]

[{ph ∈ ∗ | ∗} →^{pyk} “the set of ph in " such that " end set”] The pyk compiler, version 0.grue.20060417+ by Klaus Grue GRD-2006-08-24.UTC:08:13:50.904458 = MJD-53971.TAI:08:14:23.904458 = LGT-4663124063904458e-6