Axiom ax-inf 9661

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 9645 and inf2 9646). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9665 and omex 9666 and are based on the (nontrivial) proof of inf3 9658. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9664. Theorem inf0 9644 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9668 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 9664 requires this axiom along with Regularity ax-reg 9615 for its derivation (as Theorem axinf2 9663 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9664 instead of this one. The derivation of this axiom from ax-inf2 9664 is shown by Theorem axinf 9667.

Proofs should normally use the standard version ax-inf2 9664 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 2099	. . 3 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	4, 2	wel 2099	. . . . 5 wff 𝑧 ∈ 𝑦
6		vw	. . . . . . . 8 setvar 𝑤
7	4, 6	wel 2099	. . . . . . 7 wff 𝑧 ∈ 𝑤
8	6, 2	wel 2099	. . . . . . 7 wff 𝑤 ∈ 𝑦
9	7, 8	wa 394	. . . . . 6 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
10	9, 6	wex 1773	. . . . 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 1531	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13	3, 12	wa 394	. 2 wff (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
14	13, 2	wex 1773	1 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

This axiom is referenced by:  zfinf  9662