Axiom ax-inf 9642

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 9626 and inf2 9627). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9646 and omex 9647 and are based on the (nontrivial) proof of inf3 9639. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9645. Theorem inf0 9625 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9649 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 9645 requires this axiom along with Regularity ax-reg 9589 for its derivation (as Theorem axinf2 9644 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9645 instead of this one. The derivation of this axiom from ax-inf2 9645 is shown by Theorem axinf 9648.

Proofs should normally use the standard version ax-inf2 9645 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-inf

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	wel 2145	. . 3 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	4, 2	wel 2145	. . . . 5 wff 𝑧 ∈ 𝑦
6		vw	. . . . . . . 8 setvar 𝑤
7	4, 6	wel 2145	. . . . . . 7 wff 𝑧 ∈ 𝑤
8	6, 2	wel 2145	. . . . . . 7 wff 𝑤 ∈ 𝑦
9	7, 8	wa 401	. . . . . 6 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
10	9, 6	wex 1811	. . . . 5 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
11	5, 10	wi 4	. . . 4 wff (𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
12	11, 4	wal 1567	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13	3, 12	wa 401	. 2 wff (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
14	13, 2	wex 1811	1 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

This axiom is used by:  zfinf  9643