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Axiom ax-inf 8228
 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 8212 and inf2 8213). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8232 and omex 8233 and are based on the (nontrivial) proof of inf3 8225. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8231. Theorem inf0 8211 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8235 shows a very short way to state this axiom. The standard version of Infinity ax-inf2 8231 requires this axiom along with Regularity ax-reg 8190 for its derivation (as theorem axinf2 8230 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8231 instead of this one. The derivation of this axiom from ax-inf2 8231 is shown by theorem axinf 8234. Proofs should normally use the standard version ax-inf2 8231 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)
Assertion
Ref Expression
ax-inf 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-inf
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
2 vy . . . 4 setvar 𝑦
31, 2wel 1938 . . 3 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
54, 2wel 1938 . . . . 5 wff 𝑧𝑦
6 vw . . . . . . . 8 setvar 𝑤
74, 6wel 1938 . . . . . . 7 wff 𝑧𝑤
86, 2wel 1938 . . . . . . 7 wff 𝑤𝑦
97, 8wa 378 . . . . . 6 wff (𝑧𝑤𝑤𝑦)
109, 6wex 1692 . . . . 5 wff 𝑤(𝑧𝑤𝑤𝑦)
115, 10wi 4 . . . 4 wff (𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
1211, 4wal 1466 . . 3 wff 𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
133, 12wa 378 . 2 wff (𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
1413, 2wex 1692 1 wff 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
 Colors of variables: wff setvar class This axiom is referenced by:  zfinf  8229
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