Axiom ax-inf 9666

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 9650 and inf2 9651). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9670 and omex 9671 and are based on the (nontrivial) proof of inf3 9663. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9669. Theorem inf0 9649 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9673 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 9669 requires this axiom along with Regularity ax-reg 9613 for its derivation (as Theorem axinf2 9668 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9669 instead of this one. The derivation of this axiom from ax-inf2 9669 is shown by Theorem axinf 9672.

Proofs should normally use the standard version ax-inf2 9669 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 2148	. . 3 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	4, 2	wel 2148	. . . . 5 wff 𝑧 ∈ 𝑦
6		vw	. . . . . . . 8 setvar 𝑤
7	4, 6	wel 2148	. . . . . . 7 wff 𝑧 ∈ 𝑤
8	6, 2	wel 2148	. . . . . . 7 wff 𝑤 ∈ 𝑦
9	7, 8	wa 401	. . . . . 6 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
10	9, 6	wex 1812	. . . . 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 1568	. . 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 1812	1 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

This axiom is used by:  zfinf  9667