Axiom ax-inf 8832

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 8816 and inf2 8817). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8836 and omex 8837 and are based on the (nontrivial) proof of inf3 8829. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8835. Theorem inf0 8815 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8839 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8835 requires this axiom along with Regularity ax-reg 8786 for its derivation (as theorem axinf2 8834 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8835 instead of this one. The derivation of this axiom from ax-inf2 8835 is shown by theorem axinf 8838.

Proofs should normally use the standard version ax-inf2 8835 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 2107	. . 3 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	4, 2	wel 2107	. . . . 5 wff 𝑧 ∈ 𝑦
6		vw	. . . . . . . 8 setvar 𝑤
7	4, 6	wel 2107	. . . . . . 7 wff 𝑧 ∈ 𝑤
8	6, 2	wel 2107	. . . . . . 7 wff 𝑤 ∈ 𝑦
9	7, 8	wa 386	. . . . . 6 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
10	9, 6	wex 1823	. . . . 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 1599	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13	3, 12	wa 386	. 2 wff (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
14	13, 2	wex 1823	1 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

This axiom is referenced by:  zfinf  8833