P. 96 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ixpiunwdom 9501*	Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8862 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))
Theorem	harwdom 9502	The value of the Hartogs function at a set 𝑋 is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 9455 to prove that (har‘𝑋) is an ordinal. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity
Axiom	ax-reg 9503*	Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 9507) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 9508). A stronger version that works for proper classes is proved as zfregs 9647. (Contributed by NM, 14-Aug-1993.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	axreg2 9504*	Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	zfregcl 9505*	The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) Avoid ax-10 2142 and ax-12 2178. (Revised by TM, 31-Dec-2025.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
Theorem	zfregclOLD 9506*	Obsolete version of zfregcl 9505 as of 31-Dec-2025. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
Theorem	zfreg 9507*	The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form". Axiom Reg of [BellMachover] p. 480. There is also a "strong form", not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 9647). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	elirrv 9508	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. This is trivial to prove from zfregfr 9519 and efrirr 5603 (see elirrvALT 9520), but this proof is direct from ax-reg 9503. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9503 directly. (Revised by BTernaryTau, 27-Dec-2025.)
⊢ ¬ 𝑥 ∈ 𝑥
Theorem	elirrvOLD 9509	Obsolete version of elirrv 9508 as of 27-Dec-2025. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝑥 ∈ 𝑥
Theorem	elirr 9510	No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ¬ 𝐴 ∈ 𝐴
Theorem	elneq 9511	A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)
Theorem	nelaneq 9512	A class is not an element of and equal to a class at the same time. Variant of elneq 9511 analogously to elnotel 9525 and en2lp 9521. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)
Theorem	nelaneqOLD 9513	Obsolete version of nelaneq 9512 as of 31-Dec-2025. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)
Theorem	epinid0 9514	The membership relation and the identity relation are disjoint. Variable-free version of nelaneq 9512. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)
⊢ ( E ∩ I ) = ∅
Theorem	sucprcreg 9515	A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)
⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Theorem	ruv 9516	The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
Theorem	ruALT 9517	Alternate proof of ru 3742, simplified using (indirectly) the Axiom of Regularity ax-reg 9503. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
Theorem	disjcsn 9518	A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 4-Apr-2019.)
⊢ (𝐴 ∩ {𝐴}) = ∅
Theorem	zfregfr 9519	The membership relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
⊢ E Fr 𝐴
Theorem	elirrvALT 9520	Alternate proof of elirrv 9508, shorter but using more axioms. (Contributed by BTernaryTau, 28-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝑥 ∈ 𝑥
Theorem	en2lp 9521	No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
Theorem	elnanel 9522	Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9521 and serves as an example in the context of Godel codes, see elnanelprv 35401. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)
Theorem	cnvepnep 9523	The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9521. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.)
⊢ (◡ E ∩ E ) = ∅
Theorem	epnsym 9524	The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
⊢ ◡ E ≠ E
Theorem	elnotel 9525	A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	elnel 9526	A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
⊢ (𝐴 ∈ 𝐵 → 𝐵 ∉ 𝐴)
Theorem	en3lplem1 9527*	Lemma for en3lp 9529. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Theorem	en3lplem2 9528*	Lemma for en3lp 9529. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
Theorem	en3lp 9529	No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 44818 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)
Theorem	preleqg 9530	Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9531. (Contributed by AV, 15-Jun-2022.)
⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	preleq 9531	Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Revised by AV, 15-Jun-2022.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	preleqALT 9532	Alternate proof of preleq 9531, not based on preleqg 9530: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthreg 9533	Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 9503 (via the preleq 9531 step). See df-op 4586 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof shortened by AV, 15-Jun-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	suc11reg 9534	The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)
Theorem	dford2 9535*	Assuming ax-reg 9503, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
2.5.2  Axiom of Infinity equivalents
Theorem	inf0 9536*	Existence of ω implies our axiom of infinity ax-inf 9553. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 9553. (Contributed by NM, 15-Oct-1996.) Revised to closed form. (Revised by BJ, 20-May-2024.)
⊢ (ω ∈ 𝑉 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
Theorem	inf1 9537	Variation of Axiom of Infinity (using zfinf 9554 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))    ⇒   ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
Theorem	inf2 9538*	Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 9554 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))    ⇒   ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
Theorem	inf3lema 9539*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))
Theorem	inf3lemb 9540*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹‘∅) = ∅
Theorem	inf3lemc 9541*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹‘𝐴)))
Theorem	inf3lemd 9542*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)
Theorem	inf3lem1 9543*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))
Theorem	inf3lem2 9544*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 28-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥))
Theorem	inf3lem3 9545*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 9507. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴)))
Theorem	inf3lem4 9546*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴)))
Theorem	inf3lem5 9547*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
Theorem	inf3lem6 9548*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. (Contributed by NM, 29-Oct-1996.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
Theorem	inf3lem7 9549*	Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7899. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})    &   ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
Theorem	inf3 9550	Our Axiom of Infinity ax-inf 9553 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 9538, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 9555 and zfinf2 9557.) The main proof is provided by inf3lema 9539 through inf3lem7 9549, and this final piece eliminates the auxiliary hypothesis of inf3lem7 9549. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof. (As posted to sci.logic on 30-Oct-1996, with annotations added.) Theorem: The statement "There exists a nonempty set that is a subset of its union" implies the Axiom of Infinity. Proof: Let X be a nonempty set which is a subset of its union; the latter property is equivalent to saying that for any y in X, there exists a z in X such that y is in z. Define by finite recursion a function F:omega-->(power X) such that F_0 = 0 (See inf3lemb 9540.) F_n+1 = {y<X | y^X subset F_n} (See inf3lemc 9541.) Note: ^ means intersect, < means \in ("element of"). (Finite recursion as typically done requires the existence of omega; to avoid this we can just use transfinite recursion restricted to omega. F is a class-term that is not necessarily a set at this point.) Lemma 1. F_n subset F_n+1. (See inf3lem1 9543.) Proof: By induction: F_0 subset F_1. If y < F_n+1, then y^X subset F_n, so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2. Lemma 2. F_n =/= X. (See inf3lem2 9544.) Proof: By induction: F_0 =/= X because X is not empty. Assume F_n =/= X. Then there is a y in X that is not in F_n. By definition of X, there is a z in X that contains y. Suppose F_n+1 = X. Then z is in F_n+1, and z^X contains y, so z^X is not a subset of F_n, contrary to the definition of F_n+1. Lemma 3. F_n =/= F_n+1. (See inf3lem3 9545.) Proof: Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have F_n+1 = {y<X | y^(X-F_n) = 0}. Let q = {y<X-F_n | y^(X-F_n) = 0}. Then q subset F_n+1. Since X-F_n is not empty by Lemma 2 and q is the set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q and therefore F_n+1 have an element not in F_n. Lemma 4. F_n proper_subset F_n+1. (See inf3lem4 9546.) Proof: Lemmas 1 and 3. Lemma 5. F_m proper_subset F_n, m < n. (See inf3lem5 9547.) Proof: Fix m and use induction on n > m. Basis: F_m proper_subset F_m+1 by Lemma 4. Induction: Assume F_m proper_subset F_n. Then since F_n proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper subset. By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1. (See inf3lem6 9548.) Thus, the inverse of F is a function with range omega and domain a subset of power X, so omega exists by Replacement. (See inf3lem7 9549.) Q.E.D. (Contributed by NM, 29-Oct-1996.)
⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)    ⇒   ⊢ ω ∈ V
Theorem	infeq5i 9551	Half of infeq5 9552. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
Theorem	infeq5 9552	The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 9558.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V)
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-inf 9553*	Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set 𝑥, an infinite set 𝑦 built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 9537 and inf2 9538). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9557 and omex 9558 and are based on the (nontrivial) proof of inf3 9550. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9556. Theorem inf0 9536 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9560 shows a very short way to state this axiom. The standard version of Infinity ax-inf2 9556 requires this axiom along with Regularity ax-reg 9503 for its derivation (as Theorem axinf2 9555 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9556 instead of this one. The derivation of this axiom from ax-inf2 9556 is shown by Theorem axinf 9559. Proofs should normally use the standard version ax-inf2 9556 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
Theorem	zfinf 9554*	Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
Theorem	axinf2 9555*	A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 9553 and Regularity ax-reg 9503. This theorem should not be referenced in any proof. Instead, use ax-inf2 9556 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)
⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Axiom	ax-inf2 9556*	A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 9557 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9555 above, using our version of Infinity ax-inf 9553 and the Axiom of Regularity ax-reg 9503. We will reference ax-inf2 9556 instead of axinf2 9555 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9553 from ax-inf2 9556 is shown by Theorem axinf 9559. (Contributed by NM, 3-Nov-1996.)
⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	zfinf2 9557*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9556 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
2.6.2  Existence of omega (the set of natural numbers)
Theorem	omex 9558	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. Remark 1.21 of [Schloeder] p. 3. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 9536. A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 7818 and Fin = V (the universe of all sets) by fineqv 9168. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7829 through peano5 7833 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)
⊢ ω ∈ V
Theorem	axinf 9559*	The first version of the Axiom of Infinity ax-inf 9553 proved from the second version ax-inf2 9556. Note that we didn't use ax-reg 9503, unlike the other direction axinf2 9555. (Contributed by NM, 24-Apr-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
Theorem	inf5 9560	The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see Theorem infeq5 9552). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)
⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥
Theorem	omelon 9561	Omega is an ordinal number. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ ω ∈ On
Theorem	dfom3 9562*	The class of natural numbers ω can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Definition 1.20 of [Schloeder] p. 3. (Contributed by NM, 6-Aug-1994.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	elom3 9563*	A simplification of elom 7809 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))
Theorem	dfom4 9564*	A simplification of df-om 7807 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom5 9565	ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). Theorem 1.23 of [Schloeder] p. 4. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}
Theorem	oancom 9566	Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
⊢ (1o +o ω) ≠ (ω +o 1o)
Theorem	isfinite 9567	A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)
⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
Theorem	fict 9568	A finite set is countable (weaker version of isfinite 9567). (Contributed by Thierry Arnoux, 27-Mar-2018.)
⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω)
Theorem	nnsdom 9569	A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐴 ∈ ω → 𝐴 ≺ ω)
Theorem	omenps 9570	Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)
⊢ ω ≈ (ω ∖ {∅})
Theorem	omensuc 9571	The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
⊢ ω ≈ suc ω
Theorem	infdifsn 9572	Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Theorem	infdiffi 9573	Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)
Theorem	unbnn3 9574*	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 9201 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)
Theorem	noinfep 9575*	Using the Axiom of Regularity in the form zfregfr 9519, show that there are no infinite descending ∈-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)
2.6.3  Cantor normal form
Syntax	ccnf 9576	Extend class notation with the Cantor normal form function.
class CNF
Definition	df-cnf 9577*	Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥 ↑o 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑o 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9610 of this function in terms of df-oi 9421. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnffval 9578*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnfdm 9579*	The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Theorem	cantnfvalf 9580*	Lemma for cantnf 9608. The function appearing in cantnfval 9583 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)    ⇒   ⊢ 𝐹:ω⟶On
Theorem	cantnfs 9581	Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
Theorem	cantnfcl 9582	Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
Theorem	cantnfval 9583*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Theorem	cantnfval2 9584*	Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
Theorem	cantnfsuc 9585*	The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
Theorem	cantnfle 9586*	A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
Theorem	cantnflt 9587*	An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnflt2 9588	An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnff 9589	The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
Theorem	cantnf0 9590	The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
Theorem	cantnfrescl 9591*	A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
Theorem	cantnfres 9592*	The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    &   ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))
Theorem	cantnfp1lem1 9593*	Lemma for cantnfp1 9596. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	cantnfp1lem2 9594*	Lemma for cantnfp1 9596. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    ⇒   ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂)
Theorem	cantnfp1lem3 9595*	Lemma for cantnfp1 9596. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅)    &   ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
Theorem	cantnfp1 9596*	If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
Theorem	oemapso 9597*	The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9464). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → 𝑇 Or 𝑆)
Theorem	oemapval 9598*	Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
Theorem	oemapvali 9599*	If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
Theorem	cantnflem1a 9600*	Lemma for cantnf 9608. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))