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Axiom ax-inf 9089
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 9073 and inf2 9074). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 9093 and omex 9094 and are based on the (nontrivial) proof of inf3 9086. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 9092. Theorem inf0 9072 shows the reverse derivation of our axiom from a standard one. Theorem inf5 9096 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 9092 requires this axiom along with Regularity ax-reg 9044 for its derivation (as theorem axinf2 9091 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9092 instead of this one. The derivation of this axiom from ax-inf2 9092 is shown by theorem axinf 9095.

Proofs should normally use the standard version ax-inf2 9092 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 2115 . . 3 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
54, 2wel 2115 . . . . 5 wff 𝑧𝑦
6 vw . . . . . . . 8 setvar 𝑤
74, 6wel 2115 . . . . . . 7 wff 𝑧𝑤
86, 2wel 2115 . . . . . . 7 wff 𝑤𝑦
97, 8wa 399 . . . . . 6 wff (𝑧𝑤𝑤𝑦)
109, 6wex 1781 . . . . 5 wff 𝑤(𝑧𝑤𝑤𝑦)
115, 10wi 4 . . . 4 wff (𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
1211, 4wal 1536 . . 3 wff 𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
133, 12wa 399 . 2 wff (𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
1413, 2wex 1781 1 wff 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  zfinf  9090
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