MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-inf Structured version   Visualization version   GIF version

Axiom ax-inf 8395
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 8379 and inf2 8380). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8399 and omex 8400 and are based on the (nontrivial) proof of inf3 8392. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8398. Theorem inf0 8378 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8402 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8398 requires this axiom along with Regularity ax-reg 8357 for its derivation (as theorem axinf2 8397 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8398 instead of this one. The derivation of this axiom from ax-inf2 8398 is shown by theorem axinf 8401.

Proofs should normally use the standard version ax-inf2 8398 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 1977 . . 3 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
54, 2wel 1977 . . . . 5 wff 𝑧𝑦
6 vw . . . . . . . 8 setvar 𝑤
74, 6wel 1977 . . . . . . 7 wff 𝑧𝑤
86, 2wel 1977 . . . . . . 7 wff 𝑤𝑦
97, 8wa 382 . . . . . 6 wff (𝑧𝑤𝑤𝑦)
109, 6wex 1694 . . . . 5 wff 𝑤(𝑧𝑤𝑤𝑦)
115, 10wi 4 . . . 4 wff (𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
1211, 4wal 1472 . . 3 wff 𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
133, 12wa 382 . 2 wff (𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
1413, 2wex 1694 1 wff 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  zfinf  8396
  Copyright terms: Public domain W3C validator