P. 461 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46001-46100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unisnALT 46001	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. The User manually input on a mmj2 Proof Worksheet, without labels, all steps of unisnALT 46001 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all steps except for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15, 21, and 30). mmj2 could not find reference theorems for those five steps because the hypothesis field of each of these steps was empty and none of those steps unifies with a theorem in set.mm. Each of these five steps is a semantic variation of a theorem in set.mm and is 2-step provable. mmj2 does not have the ability to automatically generate the semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis deduction whose hypothesis is a theorem in set.mm which unifies with the theorem in the Proof Worksheet. The stepprover.c program, which invokes mmj2, has this capability. stepprover.c automatically generated steps 1, 11, 15, 21, and 30, labeled all steps, and generated the RPN proof of unisnALT 46001. Roughly speaking, stepprover.c added to the Proof Worksheet a labeled duplicate step of each non-unifying theorem for each label in a text file, labels.txt, containing a list of labels provided by the User. Upon mmj2 unification, stepprover.c identified a label for each of the five theorems which 2-step proves it. For unisnALT 46001, the label list is a list of all 1-hypothesis propositional calculus deductions in set.mm. stepproverp.c is the same as stepprover.c except that it intermittently pauses during execution, allowing the User to observe the changes to a text file caused by the execution of particular statements of the program. (Contributed by Alan Sare, 19-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ {𝐴} = 𝐴
21.45.11  Theorems with a proof in conventional notation derived from a VD proof Theorems with a proof in conventional notation automatically derived by completeusersproof.c from a Virtual Deduction User's Proof.
Theorem	notnotrALT2 46002	Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ¬ 𝜑 → 𝜑)
Theorem	sspwimpALT2 46003	If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	e2ebindALT 46004	Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 45987. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))
Theorem	ax6e2ndALT 46005*	If at least two sets exist (dtru 5404), then the same is true expressed in an alternate form similar to the form of ax6e 2412. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndVD 45983. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
Theorem	ax6e2ndeqALT 46006*	"At least two sets exist" expressed in the form of dtru 5404 is logically equivalent to the same expressed in a form similar to ax6e 2412 if dtru 5404 is false implies 𝑢 = 𝑣. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndeqVD 45984. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
Theorem	2sb5ndALT 46007*	Equivalence for double substitution 2sb5 2311 without distinct 𝑥, 𝑦 requirement. 2sb5nd 45636 is derived from 2sb5ndVD 45985. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in 2sb5ndVD 45985. (Contributed by Alan Sare, 19-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
Theorem	chordthmALT 46008*	The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA · PB and PC · PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to π. The result is proven by using chordthmlem5 27169 twice to show that PA · PB and PC · PD both equal BQ 2 − PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. Proven by David Moews on 28-Feb-2017 as chordthm 27170. https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 27170 is a Virtual Deduction User's Proof transcription of chordthm 27170. That VD User's Proof was input into completeusersproof, automatically generating this chordthmALT 46008 Metamath proof. (Contributed by Alan Sare, 19-Sep-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝑃 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝑃)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑃)    &   ⊢ (𝜑 → 𝐶 ≠ 𝑃)    &   ⊢ (𝜑 → 𝐷 ≠ 𝑃)    &   ⊢ (𝜑 → ((𝐴 − 𝑃)𝐹(𝐵 − 𝑃)) = π)    &   ⊢ (𝜑 → ((𝐶 − 𝑃)𝐹(𝐷 − 𝑃)) = π)    &   ⊢ (𝜑 → 𝑄 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄)))    &   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐶 − 𝑄)))    &   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐷 − 𝑄)))    ⇒   ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = ((abs‘(𝑃 − 𝐶)) · (abs‘(𝑃 − 𝐷))))
Theorem	isosctrlem1ALT 46009	Lemma for isosctr 27154. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 27154. As it is verified by the Metamath program, isosctrlem1ALT 46009 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 46009. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)
Theorem	iunconnlem2 46010*	The indexed union of connected overlapping subspaces sharing a common point is connected. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/iunconlem2vd.html. As it is verified by the Metamath program, iunconnlem2 46010 verifies https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 46010. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 ↔ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣)))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)    ⇒   ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn)
Theorem	iunconnALT 46011*	The indexed union of connected overlapping subspaces sharing a common point is connected. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/iunconaltvd.html. As it is verified by the Metamath program, iunconnALT 46011 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 46011. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)    ⇒   ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn)
Theorem	sineq0ALT 46012	A complex number whose sine is zero is an integer multiple of π. The Virtual Deduction form of the proof is https://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 46012. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 26857. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/sineq0altro.html 26857 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) = 0 ↔ (𝐴 / π) ∈ ℤ))
21.46  Mathbox for Eric Schmidt
21.46.1  Miscellany
Theorem	rspesbcd 46013*	Restricted quantifier version of spesbcd 3829. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rext0 46014*	Nonempty existential quantification of a theorem is true. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝜑    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)
Theorem	cocanss2 46015	Cancellation law for composition. Suggested by BJ. (Contributed by Eric Schmidt, 30-Sep-2026.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Rel 𝐻)    &   ⊢ (𝜑 → dom 𝐻 ⊆ ran 𝐹)    ⇒   ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) ↔ 𝐻 ⊆ 𝐾))
Theorem	cocan2g 46016	Cancellation law for composition. A version of cocan2 7300 with weaker hypotheses. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Rel 𝐻)    &   ⊢ (𝜑 → dom 𝐻 ⊆ ran 𝐹)    &   ⊢ (𝜑 → Rel 𝐾)    &   ⊢ (𝜑 → dom 𝐾 ⊆ ran 𝐹)    ⇒   ⊢ (𝜑 → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))
Theorem	cocanss1 46017	Cancellation law for composition. Suggested by BJ. (Contributed by Eric Schmidt, 30-Sep-2026.)
⊢ (𝜑 → Fun ◡𝐹)    &   ⊢ (𝜑 → Rel 𝐻)    &   ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ 𝐻 ⊆ 𝐾))
Theorem	cocan1g 46018	Cancellation law for composition. A version of cocan1 7299 with weaker hypotheses. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝜑 → Fun ◡𝐹)    &   ⊢ (𝜑 → Rel 𝐻)    &   ⊢ (𝜑 → ran 𝐻 ⊆ dom 𝐹)    &   ⊢ (𝜑 → Rel 𝐾)    &   ⊢ (𝜑 → ran 𝐾 ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))
Theorem	dmstructnn 46019	The domain of an extensible structure is a subset of ℕ. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 Struct 𝑋 → dom 𝐹 ⊆ ℕ)
Theorem	dmstructfi 46020	The domain of an extensible structure is finite. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 Struct 𝑋 → dom 𝐹 ∈ Fin)
Theorem	rnstructfi 46021	The range of an extensible structure is finite. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 Struct 𝑋 → ran 𝐹 ∈ Fin)
21.46.2  Study of dfbi1ALT
Theorem	dfbi1ALTa 46022	Version of dfbi1ALT 217 using ⊤ for step 2 and shortened using a1i 11, a2i 15, and con4i 115. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
Theorem	simprimi 46023	Inference associated with simprim 167. Proved exactly as step 11 is obtained from step 4 in dfbi1ALTa 46022. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ¬ (𝜑 → ¬ 𝜓)    ⇒   ⊢ 𝜓
Theorem	dfbi1ALTb 46024	Further shorten dfbi1ALTa 46022 using simprimi 46023. (Contributed by Eric Schmidt, 22-Oct-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
21.46.3  Relation-preserving functions
Syntax	wrelp 46025	Extend the definition of a wff to include the relation-preserving property. (Contributed by Eric Schmidt, 11-Oct-2025.)
wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
Definition	df-relp 46026*	Define the relation-preserving predicate. This is a viable notion of "homomorphism" corresponding to df-isom 6547. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
Theorem	relpeq1 46027	Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐻 = 𝐺 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐺 RelPres 𝑅, 𝑆(𝐴, 𝐵)))
Theorem	relpeq2 46028	Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝑅 = 𝑇 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑇, 𝑆(𝐴, 𝐵)))
Theorem	relpeq3 46029	Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝑆 = 𝑇 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑇(𝐴, 𝐵)))
Theorem	relpeq4 46030	Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐴 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐶, 𝐵)))
Theorem	relpeq5 46031	Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐵 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐶)))
Theorem	nfrelp 46032	Bound-variable hypothesis builder for a relation-preserving function. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ Ⅎ𝑥𝐻    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
Theorem	relpf 46033	A relation-preserving function is a function. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
Theorem	relprel 46034	A relation-preserving function preserves the relation. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 → (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	relpmin 46035	A preimage of a minimal element under a relation-preserving function is minimal. Essentially one half of isomin 7345. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
Theorem	relpfrlem 46036*	Lemma for relpfr 46037. Proved without using the Axiom of Replacement. This is isofrlem 7348 with weaker hypotheses. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝜑 → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	relpfr 46037	If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7350 for a bidirectional statement. A more general version of Lemma I.9.9 of [Kunen2] p. 47. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
21.46.4  Orbits
Theorem	orbitex 46038	Orbits exist. Given a set 𝐴 and a function 𝐹, the orbit of 𝐴 under 𝐹 is the smallest set 𝑍 such that 𝐴 ∈ 𝑍 and 𝑍 is closed under 𝐹. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (rec(𝐹, 𝐴) “ ω) ∈ V
Theorem	orbitinit 46039	A set is contained in its orbit. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (rec(𝐹, 𝐴) “ ω))
Theorem	orbitcl 46040	The orbit under a function is closed under the function. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))
Theorem	orbitclmpt 46041	Version of orbitcl 46040 using maps-to notation. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝑍 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) “ ω)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ 𝑍 ∧ 𝐷 ∈ 𝑉) → 𝐷 ∈ 𝑍)
21.46.5  Well-founded sets
Theorem	trwf 46042	The class of well-founded sets is transitive. (Contributed by Eric Schmidt, 9-Sep-2025.)
⊢ Tr ∪ (𝑅1 “ On)
Theorem	rankrelp 46043	The rank function preserves ∈. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ rank RelPres E , E (∪ (𝑅1 “ On), On)
Theorem	wffr 46044	The class of well-founded sets is well-founded. Lemma I.9.24(2) of [Kunen2] p. 53. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ E Fr ∪ (𝑅1 “ On)
Theorem	trfr 46045	A transitive class well-founded by ∈ is a subclass of the class of well-founded sets. Part of Lemma I.9.21 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
Theorem	tcfr 46046	A set is well-founded if and only if its transitive closure is well-founded by ∈. This characterization of well-founded sets is that in Definition I.9.20 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))
Theorem	xpwf 46047	The Cartesian product of two well-founded sets is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))
Theorem	dmwf 46048	The domain of a well-founded set is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → dom 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	rnwf 46049	The range of a well-founded set is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ran 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	relwf 46050	A relation is a well-founded set iff its domain and range are. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))
21.46.6  Absoluteness in transitive models
Theorem	ralabso 46051*	Simplification of restricted quantification in a transitive class. When 𝜑 is quantifier-free, this shows that the formula ∀𝑥 ∈ 𝑦𝜑 is absolute for transitive models, which is a particular case of Lemma I.16.2 of [Kunen2] p. 95. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜑)))
Theorem	rexabso 46052*	Simplification of restricted quantification in a transitive class. When 𝜑 is quantifier-free, this shows that the formula ∃𝑥 ∈ 𝑦𝜑 is absolute for transitive models, which is a particular case of Lemma I.16.2 of [Kunen2] p. 95. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	ralabsod 46053*	Deduction form of ralabso 46051. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))
Theorem	rexabsod 46054*	Deduction form of rexabso 46052. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))
Theorem	ralabsobidv 46055*	Formula-building lemma for proving absoluteness results. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜒)))
Theorem	rexabsobidv 46056*	Formula-building lemma for proving absoluteness results. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜒)))
Theorem	ssabso 46057*	The notion "𝑥 is a subset of 𝑦 " is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
Theorem	disjabso 46058*	Disjointness is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
Theorem	n0abso 46059*	Nonemptiness is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))
21.46.7  Lemmas for showing axioms hold in models
Theorem	traxext 46060*	A transitive class models the Axiom of Extensionality ax-ext 2732. Lemma II.2.4(1) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 11-Sep-2025.)
⊢ (Tr 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Theorem	modelaxreplem1 46061*	Lemma for modelaxrep 46064. We show that 𝑀 is closed under taking subsets. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ 𝐴 ⊆ 𝑥    ⇒   ⊢ (𝜓 → 𝐴 ∈ 𝑀)
Theorem	modelaxreplem2 46062*	Lemma for modelaxrep 46064. We define a class 𝐹 and show that the antecedent of Replacement implies that 𝐹 is a function. We use Replacement (in the form of funex 7225) to show that 𝐹 exists. Then we show that, under our hypotheses, the range of 𝐹 is a member of 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ Ⅎ𝑤𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ Ⅎ𝑧𝐹    &   ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}    &   ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))    ⇒   ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)
Theorem	modelaxreplem3 46063*	Lemma for modelaxrep 46064. We show that the consequent of Replacement is satisfied with ran 𝐹 as the value of 𝑦. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ Ⅎ𝑤𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ Ⅎ𝑧𝐹    &   ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}    &   ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))    ⇒   ⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	modelaxrep 46064*	Conditions which guarantee that a class models the Axiom of Replacement ax-rep 5231. Similar to Lemma II.2.4(6) of [Kunen2] p. 111. The first two hypotheses are those in Kunen. The reason for the third hypothesis that our version of Replacement is different from Kunen's (which is zfrep6 5241). If we assumed Regularity, we could eliminate this extra hypothesis, since under Regularity, the empty set is a member of every non-empty transitive class. Note that, to obtain the relativization of an instance of Replacement to 𝑀, the formula ∀𝑦𝜑 would need to be replaced with ∀𝑦 ∈ 𝑀𝜒, where 𝜒 is 𝜑 with all quantifiers relativized to 𝑀. However, we can obtain this by using 𝑦 ∈ 𝑀 ∧ 𝜒 for 𝜑 in this theorem, so it does establish that all instances of Replacement hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → Tr 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    ⇒   ⊢ (𝜓 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
Theorem	ssclaxsep 46065*	A class that is closed under subsets models the Axiom of Separation ax-sep 5248. Lemma II.2.4(3) of [Kunen2] p. 111. Note that, to obtain the relativization of an instance of Separation to 𝑀, the formula 𝜑 would need to be replaced with its relativization to 𝑀. However, this new formula is a valid substitution for 𝜑, so this theorem does establish that all instances of Separation hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (∀𝑧 ∈ 𝑀 𝒫 𝑧 ⊆ 𝑀 → ∀𝑧 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
Theorem	0elaxnul 46066*	A class that contains the empty set models the Null Set Axiom ax-nul 5259. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (∅ ∈ 𝑀 → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)
Theorem	pwclaxpow 46067*	Suppose 𝑀 is a transitive class that is closed under power sets intersected with 𝑀. Then, 𝑀 models the Axiom of Power Sets ax-pow 5326. One direction of Lemma II.2.8 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝒫 𝑥 ∩ 𝑀) ∈ 𝑀) → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	prclaxpr 46068*	A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5390. Lemma II.2.4(4) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
Theorem	uniclaxun 46069*	A class that is closed under the union operation models the Axiom of Union ax-un 7751. Lemma II.2.4(5) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 1-Oct-2025.)
⊢ (∀𝑥 ∈ 𝑀 ∪ 𝑥 ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	sswfaxreg 46070*	A subclass of the class of well-founded sets models the Axiom of Regularity ax-reg 9589. Lemma II.2.4(2) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝑀 ⊆ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 𝑦 ∈ 𝑥 → ∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))))
Theorem	omssaxinf2 46071*	A class that contains all ordinals up to and including ω models the Axiom of Infinity ax-inf2 9645. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9642. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((ω ⊆ 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
Theorem	omelaxinf2 46072*	A transitive class that contains ω models the Axiom of Infinity ax-inf2 9645. Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Infinity uses the defined notions ∅ and suc, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9642. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
Theorem	dfac5prim 46073*	dfac5 10210 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (CHOICE ↔ ∀𝑥((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
Theorem	ac8prim 46074*	ac8 10573 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
Theorem	modelac8prim 46075*	If 𝑀 is a transitive class, then the following are equivalent. (1) Every nonempty set 𝑥 ∈ 𝑀 of pairwise disjoint nonempty sets has a choice set in 𝑀. (2) The class 𝑀 models the Axiom of Choice, in the form ac8prim 46074. Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including ∅ and ∩, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (Tr 𝑀 → (∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))
21.46.8  The class of well-founded sets is a model for ZFC
Theorem	wfaxext 46076*	The class of well-founded sets models the Axiom of Extensionality ax-ext 2732. Part of Corollary II.2.5 of [Kunen2] p. 112. This is the first of a series of theorems showing that all the axioms of ZFC hold in the class of well-founded sets, which we here denote by 𝑊. More precisely, for each axiom of ZFC, we obtain a provable statement if we restrict all quantifiers to 𝑊 (including implicit universal quantifiers on free variables). None of these proofs use the Axiom of Regularity. In particular, the Axiom of Regularity itself is proved to hold in 𝑊 without using Regularity. Further, the Axiom of Choice is used only in the proof that Choice holds in 𝑊. This has the consequence that any theorem of ZF (possibly proved using Regularity) can be proved, without using Regularity, to hold in 𝑊. This gives us a relative consistency result: If ZF without Regularity is consistent, so is ZF itself. Similarly, if ZFC without Regularity is consistent, so is ZFC itself. These consistency results are metatheorems and are part of Theorem II.2.13 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 11-Sep-2025.) (Revised by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 (∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	wfaxrep 46077*	The class of well-founded sets models the Axiom of Replacement ax-rep 5231. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in ∀𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	wfaxsep 46078*	The class of well-founded sets models the Axiom of Separation ax-sep 5248. Actually, our statement is stronger, since it is an instance of Separation only when all quantifiers in 𝜑 are relativized to 𝑊. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	wfaxnul 46079*	The class of well-founded sets models the Null Set Axiom ax-nul 5259. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∃𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 ¬ 𝑦 ∈ 𝑥
Theorem	wfaxpow 46080*	The class of well-founded sets models the Axioms of Power Sets. Part of Corollary II.2.9 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	wfaxpr 46081*	The class of well-founded sets models the Axiom of Pairing ax-pr 5390. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 ∃𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	wfaxun 46082*	The class of well-founded sets models the Axiom of Union ax-un 7751. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∃𝑤 ∈ 𝑊 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	wfaxreg 46083*	The class of well-founded sets models the Axiom of Regularity ax-reg 9589. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	wfaxinf2 46084*	The class of well-founded sets models the Axiom of Infinity ax-inf2 9645. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∃𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	wfac8prim 46085*	The class of well-founded sets 𝑊 models the Axiom of Choice. Since the previous theorems show that all the ZF axioms hold in 𝑊, we may use any statement that ZF proves is equivalent to Choice to prove this. We use ac8prim 46074. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
21.46.9  Permutation models
Theorem	brpermmodel 46086	The membership relation in a permutation model. We use a permutation 𝐹 of the universe to define a relation 𝑅 that serves as the membership relation in our model. The conclusion of this theorem is Definition II.9.1 of [Kunen2] p. 148. All the axioms of ZFC except for Regularity hold in permutation models, and Regularity will be false if 𝐹 is chosen appropriately. Thus, permutation models can be used to show that Regularity does not follow from the other axioms (with the usual proviso that the axioms are consistent). (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))
Theorem	brpermmodelcnv 46087	Ordinary membership expressed in terms of the permutation model's membership relation. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅(◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)
Theorem	permaxext 46088*	The Axiom of Extensionality ax-ext 2732 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
Theorem	permaxrep 46089*	The Axiom of Replacement ax-rep 5231 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. Note that, to prove that an instance of Replacement holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Replacement holds. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
Theorem	permaxsep 46090*	The Axiom of Separation ax-sep 5248 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))
Theorem	permaxnul 46091*	The Null Set Axiom ax-nul 5259 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑥∀𝑦 ¬ 𝑦𝑅𝑥
Theorem	permaxpow 46092*	The Axiom of Power Sets ax-pow 5326 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤𝑅𝑧 → 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
Theorem	permaxpr 46093*	The Axiom of Pairing ax-pr 5390 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧)
Theorem	permaxun 46094*	The Axiom of Union ax-un 7751 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
Theorem	permaxinf2lem 46095*	Lemma for permaxinf2 46096. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝑍 = (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω)    ⇒   ⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	permaxinf2 46096*	The Axiom of Infinity ax-inf2 9645 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	permac8prim 46097*	The Axiom of Choice ac8prim 46074 holds in permutation models. Part of Exercise II.9.3 of [Kunen2] p. 149. Note that ax-ac 10540 requires Regularity for its derivation from the usual Axiom of Choice and does not necessarily hold in permutation models. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
Theorem	nregmodelf1o 46098	Define a permutation 𝐹 used to produce a model in which ax-reg 9589 is false. The permutation swaps ∅ and {∅} and leaves the rest of 𝑉 fixed. This is an example given after Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    ⇒   ⊢ 𝐹:V–1-1-onto→V
Theorem	nregmodellem 46099	Lemma for nregmodel 46100. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (𝑥𝑅∅ ↔ 𝑥 ∈ {∅})
Theorem	nregmodel 46100*	The Axiom of Regularity ax-reg 9589 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 46088 through permac8prim 46097), we can conclude that Regularity does not follow from those axioms, assuming ZFC is consistent. (If we could prove Regularity from the other axioms, we could prove it in the permutation model and thus obtain a contradiction with this theorem.) Since we also know that Regularity is consistent with the other axioms (wfaxext 46076 through wfac8prim 46085), Regularity is neither provable nor disprovable from the other axioms; i.e., it is independent of them. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))