P. 357 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xoromon 35601	ω is either an ordinal set or the proper class of all ordinal sets, but not both. This is a stronger version of omon 7878. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (ω ∈ On ⊻ ω = On)
Theorem	fissorduni 35602	The union (supremum) of a finite set of ordinals less than a nonzero ordinal class is an element of that ordinal class. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ 𝐴 ∈ 𝐵)
Theorem	ordtypeon 35603	A proper class with a set-like well-ordering is isomorphic to the proper class of all ordinal numbers. (Contributed by BTernaryTau, 9-Jun-2026.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ¬ 𝐴 ∈ V) → 𝐹 Isom E , 𝑅 (On, 𝐴))
Theorem	fnrelpredd 35604*	A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
Theorem	cardpred 35605	The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))
Theorem	nummin 35606*	Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)
⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
Theorem	1enumen 35607*	The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets. The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10174 for context on disjoint union as a representation of cardinal addition. This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumcard 35608 for a version that uses the card function, 1enumkard 35706 for a version that uses the kard function , and 1enum 35726 for a version that uses an explicit sum of complex number 1s. (Contributed by BTernaryTau, 26-Jun-2026.)
⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))
Theorem	1enumcard 35608*	The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets. The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10174 for context on disjoint union as a representation of cardinal addition. This theorem does not depend on AC, but it is only meaningful for numerable sets. See 1enumen 35607 and 1enumkard 35706 for versions that are meaningful for non-numerable sets, and see 1enum 35726 for a version that uses an explicit sum of complex number 1s. (Contributed by BTernaryTau, 26-Jun-2026.)
⊢ (𝐴 ∈ V → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))
Theorem	r11 35609	Value of the cumulative hierarchy of sets function at 1o. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝑅1‘1o) = 1o
Theorem	r12 35610	Value of the cumulative hierarchy of sets function at 2o. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅1‘2o) = 2o
Theorem	r1wf 35611	Each stage in the cumulative hierarchy is well-founded. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On)
Theorem	elwf 35612	An element of a well-founded set is well-founded. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On))
Theorem	r1elcl 35613	Each set of the cumulative hierarchy is closed under membership. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵))
Theorem	rankval2b 35614*	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9804 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})
Theorem	rankval4b 35615*	The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. This variant of rankval4 9853 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
Theorem	onrankid 35616	The rank of an ordinal number is itself. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ On ↔ (rank‘𝐴) = 𝐴)
Theorem	rankfilimbi 35617*	If all elements in a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
Theorem	rankfilimb 35618*	The rank of a finite well-founded set is less than a limit ordinal iff the ranks of all of its elements are less than that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
Theorem	r1filimi 35619*	If all elements in a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
Theorem	r1filim 35620*	A finite set appears in the cumulative hierarchy prior to a limit ordinal iff all of its elements appear in the cumulative hierarchy prior to that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ Lim 𝐵) → (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
Theorem	r1omfi 35621	Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅1 “ ω) ⊆ Fin
Theorem	r1omhf 35622*	A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ ω) ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ ω)))
Theorem	r1ssel 35623	A set is a subset of the value of the cumulative hierarchy of sets function iff it is an element of the value at the successor. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵)))
Theorem	axnulALT3 35624*	Alternate proof of axnul 5266, proved from propositional calculus, ax-gen 1828, ax-4 1842, ax-5 1943, and ax-inf2 9624. (Contributed by BTernaryTau, 22-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axprALT2 35625*	Alternate proof of axpr 5396, proved from predicate calculus, ax-rep 5236, and ax-inf2 9624. (Contributed by BTernaryTau, 26-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	r1omfv 35626	Value of the cumulative hierarchy of sets function at ω. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
Theorem	rankfo 35627	The rank function maps the universe onto the ordinals. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ rank:V–onto→On
Theorem	rankfn 35628	The rank function is a function on the universe. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ rank Fn V
Theorem	trssfir1om 35629	If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
Theorem	r1omhfb 35630*	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
Theorem	scotteqi 35631	Equality theorem for the Scott operation. Inference form of scotteq 9874. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ Scott 𝐴 = Scott 𝐵
Theorem	elscott 35632*	Membership in a Scott's trick set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))
Theorem	dfscott2 35633*	Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
Theorem	dfscott3 35634	Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ Scott 𝐴 = (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴)))
Theorem	elscott2 35635	Membership in a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ (rank‘𝐴) = ∩ (rank “ 𝐵)))
Theorem	elscottrank 35636	The rank of an element in a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 → (rank‘𝐴) = ∩ (rank “ 𝐵))
Theorem	elscottrankeq 35637	Elements in a Scott's trick set have the same rank. (Contributed by BTernaryTau, 9-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐶 ∧ 𝐵 ∈ Scott 𝐶) → (rank‘𝐴) = (rank‘𝐵))
Theorem	elscottrankss 35638	Relationship between the ranks of an element in a Scott's trick set and an element in the input set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
Theorem	scottrankeqel 35639	If a member of the input set has the same rank as a member of the Scott's trick set, then it is also a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)
Theorem	nelscottrankgt 35640	If a member of the input set is not a member of the Scott's trick set, then its rank is greater than the rank of a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))
Theorem	scottsn 35641	Applying Scott's trick to a singleton leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ Scott {𝐴} = {𝐴}
Theorem	scott0bOLD 35642	Obsolete version of scott0b 9880 as of 18-Jul-2026. (Contributed by BTernaryTau, 3-Jul-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅)
Theorem	rankscott 35643	The rank of a nonempty Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ (𝐴 ≠ ∅ → (rank‘Scott 𝐴) = suc ∩ (rank “ 𝐴))
Theorem	rankscottu 35644	An upper bound on the rank of a Scott's trick set. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
Theorem	scottssr1 35645	Relationship between a Scott's trick set and the cumulative hierarchy. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ 𝐵 → Scott 𝐵 ⊆ (𝑅1‘suc (rank‘𝐴)))
Theorem	acnum 35646	The Axiom of Choice implies that any set is numerable. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (CHOICE → (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card))
21.5.2.1  Finitism
Theorem	prcinf 35647*	Any proper class is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. This proof holds regardless of whether the Axiom of Infinity is accepted or negated. (Contributed by BTernaryTau, 22-Jun-2025.)
⊢ (¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
Theorem	fineqvrep 35648*	If all sets are finite, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
Theorem	fineqvpow 35649*	If all sets are finite, then the Axiom of Power Sets becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
⊢ (Fin = V → ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	fineqvac 35650	If all sets are finite, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5236 and ax-pow 5334, see fineqvacALT 35651. (Contributed by BTernaryTau, 21-Sep-2024.)
⊢ (Fin = V → CHOICE)
Theorem	fineqvacALT 35651	Shorter proof of fineqvac 35650 using ax-rep 5236 and ax-pow 5334. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Fin = V → CHOICE)
Theorem	fineqvomon 35652	If all sets are finite, then the class of all natural numbers equals the proper class of all ordinal numbers. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V → ω = On)
Theorem	fineqvomonb 35653	All sets are finite iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (Fin = V ↔ ω = On)
Theorem	omprcomonb 35654	The class of all finite ordinals is a proper class iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (¬ ω ∈ V ↔ ω = On)
Theorem	fineqvnttrclselem1 35655*	Lemma for fineqvnttrclse 35658. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
Theorem	fineqvnttrclselem2 35656*	Lemma for fineqvnttrclse 35658. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)
Theorem	fineqvnttrclselem3 35657*	Lemma for fineqvnttrclse 35658. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   ⊢ 𝐴 = ω    &   ⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → ∀𝑎 ∈ suc 𝑁(𝐹‘𝑎)𝑅(𝐹‘suc 𝑎))
Theorem	fineqvnttrclse 35658*	A counterexample demonstrating that ttrclse 9710 does not hold when all sets are finite. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   ⊢ 𝐴 = ω    ⇒   ⊢ (Fin = V → (𝑅 Se 𝐴 ∧ ¬ t++(𝑅 ↾ 𝐴) Se 𝐴))
Theorem	fineqvinfep 35659*	A counterexample demonstrating that tz9.1 9712 does not hold when all sets are finite and an infinite descending ∈-chain exists. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ 𝐴 = {(𝐹‘∅)}    ⇒   ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21.5.2.2  Introduce ax-regs
Axiom	ax-regs 35660*	A strong version of the Axiom of Regularity. It states that if there exists a set with property 𝜑, then there must exist a set with property 𝜑 such that none of its elements have property 𝜑. This axiom can be derived from the axioms of ZF set theory as shown in axregs 35673, but this derivation relies on ax-inf2 9624 and is thus not possible in a finitist context. (Contributed by BTernaryTau, 29-Dec-2025.)
⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
Theorem	axreg 35661*	Derivation of ax-reg 9568 from ax-regs 35660 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35660 is a stronger version of ax-reg 9568. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	axregscl 35662*	A version of ax-regs 35660 with a class variable instead of a wff variable. Axiom D in Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (1940), p. 6. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
Theorem	axregszf 35663*	Derivation of zfregs 9715 using ax-regs 35660. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	setindregs 35664*	Set (epsilon) induction. This version of setind 9730 replaces zfregs 9715 with axregszf 35663. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setinds2regs 35665*	Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	noinfepfnregs 35666*	There are no infinite descending ∈-chains, proven using ax-regs 35660. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
Theorem	noinfepregs 35667*	There are no infinite descending ∈-chains, proven using ax-regs 35660. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)
Theorem	tz9.1regs 35668*	Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9712 depends on ax-regs 35660 instead of ax-reg 9568 and ax-inf2 9624. This suggests a possible answer to the third question posed in tz9.1 9712, namely that the missing property is that countably infinite classes must obey regularity. In ZF set theory we can prove this by showing that countably infinite classes are sets and thus ax-reg 9568 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35660 is required since countably infinite classes are proper classes. A related candidate for the missing property is the non-existence of infinite descending ∈-chains, proven as noinfep 9643 using ax-reg 9568 and ax-inf2 9624 and as noinfepregs 35667 using ax-regs 35660. If all sets are finite, then the existence of such a chain implies there is a set which does not have a transitive closure, as shown in fineqvinfep 35659. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
Theorem	unir1regs 35669	The cumulative hierarchy of sets covers the universe. This version of unir1 9799 replaces setind 9730 with setindregs 35664. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅1 “ On) = V
Theorem	trssfir1omregs 35670	If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfir1om 35629 replaces setinds2 9734 with setinds2regs 35665. (Contributed by BTernaryTau, 20-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
Theorem	r1omhfbregs 35671*	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of r1omhfb 35630 replaces setinds2 9734 with setinds2regs 35665 and trssfir1om 35629 with trssfir1omregs 35670. (Contributed by BTernaryTau, 21-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
Theorem	fineqvr1ombregs 35672	All sets are finite iff all sets are hereditarily finite. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V ↔ ∪ (𝑅1 “ ω) = V)
21.5.2.3  Derive ax-regs
Theorem	axregs 35673*	Derivation of ax-regs 35660 from the axioms of ZF set theory. (Contributed by BTernaryTau, 29-Dec-2025.)
⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
21.5.2.4  ZFC axioms with reduced distinct variable conditions
Theorem	axsepg2 35674*	A generalization of ax-sep 5255 in which 𝑥 and 𝑧 need not be distinct. This theorem scheme bundles ax-sep 5255 with the degenerate instance ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) which is satisfied by the existence of the empty set. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 21-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg3 35675*	A generalization of ax-sep 5255 in which 𝑦 and 𝑧 need not be distinct. This theorem scheme bundles ax-sep 5255 with the degenerate instance ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) which is satisfied by the existence of the empty set. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg3ALT 35676*	Alternate proof of axsepg3 35675, derived directly from ax-sep 5255 with no additional set theory axioms. (Contributed by BTernaryTau, 3-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg4 35677*	A generalization of ax-sep 5255 that combines axsepg 5256 and axsepg2 35674 into a single theorem scheme. Unlike ax-sep 5255, this scheme lacks a distinct variable condition for 𝜑 and 𝑧 as well as for 𝑥 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg5 35678*	A generalization of ax-sep 5255 that combines axsepg 5256, axsepg2 35674, and axsepg3 35675 into a single theorem scheme. Unlike ax-sep 5255, this scheme lacks a distinct variable condition for 𝜑 and 𝑧, for 𝑥 and 𝑧, and for 𝑦 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axnulg 35679	A generalization of ax-nul 5267 in which 𝑥 and 𝑦 need not be distinct. This theorem scheme bundles ax-nul 5267 with the degenerate instance ∃𝑥∀𝑥¬ 𝑥 ∈ 𝑥 which is satisfied by elirrv 9573. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axpowg 35680*	A generalization of ax-pow 5334 that combines it and zfpow 5335 into a single theorem scheme. Unlike ax-pow 5334, this scheme lacks a distinct variable condition for 𝑦 and 𝑤. (Contributed by BTernaryTau, 26-May-2026.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	axpowg2 35681*	A generalization of ax-pow 5334 in which 𝑥 and 𝑤 need not be distinct. This theorem scheme bundles ax-pow 5334 with the degenerate instance ∃𝑦∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑥) → 𝑧 ∈ 𝑦) which is satisfied by the existence of a set that contains all empty sets (see axprlem1 5392). Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 26-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	axpowg3 35682*	A generalization of ax-pow 5334 that combines axpowg 35680 and axpowg2 35681 into a single theorem scheme. Unlike ax-pow 5334, this scheme lacks a distinct variable condition for 𝑦 and 𝑤 as well as for 𝑥 and 𝑤. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by BTernaryTau, 26-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
21.5.2.5  Cardinality without the Axiom of Choice
Syntax	ckard 35683	Extend class definition to include the alternative cardinal size function.
class kard
Definition	df-kard 35684*	Define the alternative cardinal number function. Under this definition, the cardinal number of a set is the set of all sets equinumerous to it and having the least possible rank. Definition of [Enderton] p. 222. See kardval 35686 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35691. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 9948. (Contributed by BTernaryTau, 2-Jul-2026.)
⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
Theorem	kardfn 35685	The kard class is a function on the universe. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ kard Fn V
Theorem	kardval 35686*	The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval2 35687. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘𝐴) = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
Theorem	kardval2 35687*	The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval 35686. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘𝐴) = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Theorem	kard0 35688	The kard cardinality of the empty set is the singleton of the empty set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘∅) = {∅}
Theorem	elkarden 35689	Any member of the kard cardinal number of a set is equinumerous to the set. Contrast with cardne 9974 for card cardinals. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ (kard‘𝐵) → 𝐴 ≈ 𝐵)
Theorem	kardeq0 35690	Applying kard to a class yields the empty set iff the class is a proper class. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ V)
Theorem	kardeng 35691	Two sets are equinumerous iff their kard cardinal numbers are equal. Unlike carden 10563, this theorem does not depend on the Axiom of Choice, but it does depend on the Axiom of Regularity and the Axiom of Infinity. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → ((kard‘𝐴) = (kard‘𝐵) ↔ 𝐴 ≈ 𝐵))
Theorem	kardenir 35692	If two sets are equinumerous, then their kard cardinal numbers are equal. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ≈ 𝐵 → (kard‘𝐴) = (kard‘𝐵))
Theorem	kard0b 35693	The empty set is the only set with cardinality zero. This is the kard version of cardeq0 10564. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = (kard‘∅) ↔ 𝐴 = ∅)
Theorem	kardsn 35694	A singleton has cardinality one. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → (kard‘{𝐴}) = (kard‘1o))
Theorem	karddom 35695*	One set dominates another iff an element in its kard cardinality dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≼ 𝑦)
Theorem	kardsdom 35696*	One set strictly dominates another iff an element in its kard cardinality strictly dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 6-Jul-2026.)
⊢ (𝐴 ≺ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≺ 𝑦)
Theorem	kardexen 35697*	One set is equinumerous to another iff an element in its kard cardinality is equinumerous to an element in the second set's kard cardinality. See kardeng 35691 for a version with equality of cardinals. (Contributed by BTernaryTau, 7-Jul-2026.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≈ 𝑦)
Theorem	kardcard2a 35698	If two sets have equal nonzero card cardinalities, then they have equal kard cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (kard‘𝐴) = (kard‘𝐵))
Theorem	kardcard2b 35699	If two sets have equal kard cardinalities, then they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = (kard‘𝐵) → (card‘𝐴) = (card‘𝐵))
Theorem	kardcard2 35700	Two numerable sets have equal kard cardinalities iff they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵)))