Axiom ax-inf

Description: 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 and inf2 ). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 and omex and are based on the (nontrivial) proof of inf3 . This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 . Theorem inf0 shows the reverse derivation of our axiom from a standard one. Theorem inf5 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 requires this axiom along with Regularity ax-reg for its derivation (as Theorem axinf2 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 instead of this one. The derivation of this axiom from ax-inf2 is shown by Theorem axinf .

Proofs should normally use the standard version ax-inf2 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993)

	Ref	Expression
Assertion	ax-inf	⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vy	⊢ 𝑦
1		vx	⊢ 𝑥
2	1	cv	⊢ 𝑥
3	0	cv	⊢ 𝑦
4	2 3	wcel	⊢ 𝑥 ∈ 𝑦
5		vz	⊢ 𝑧
6	5	cv	⊢ 𝑧
7	6 3	wcel	⊢ 𝑧 ∈ 𝑦
8		vw	⊢ 𝑤
9	8	cv	⊢ 𝑤
10	6 9	wcel	⊢ 𝑧 ∈ 𝑤
11	9 3	wcel	⊢ 𝑤 ∈ 𝑦
12	10 11	wa	⊢ ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 )
13	12 8	wex	⊢ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 )
14	7 13	wi	⊢ ( 𝑧 ∈ 𝑦 → ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) )
15	14 5	wal	⊢ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) )
16	4 15	wa	⊢ ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) ) )
17	16 0	wex	⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) ) )