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Axiom ax-inf2 8482
Description: A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 8483 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8481 above, using our version of Infinity ax-inf 8479 and the Axiom of Regularity ax-reg 8441. We will reference ax-inf2 8482 instead of axinf2 8481 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8479 from ax-inf2 8482 is shown by theorem axinf 8485. (Contributed by NM, 3-Nov-1996.)
Assertion
Ref Expression
ax-inf2 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-inf2
StepHypRef Expression
1 vy . . . . . 6 setvar 𝑦
2 vx . . . . . 6 setvar 𝑥
31, 2wel 1988 . . . . 5 wff 𝑦𝑥
4 vz . . . . . . . 8 setvar 𝑧
54, 1wel 1988 . . . . . . 7 wff 𝑧𝑦
65wn 3 . . . . . 6 wff ¬ 𝑧𝑦
76, 4wal 1478 . . . . 5 wff 𝑧 ¬ 𝑧𝑦
83, 7wa 384 . . . 4 wff (𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦)
98, 1wex 1701 . . 3 wff 𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦)
104, 2wel 1988 . . . . . . 7 wff 𝑧𝑥
11 vw . . . . . . . . . 10 setvar 𝑤
1211, 4wel 1988 . . . . . . . . 9 wff 𝑤𝑧
1311, 1wel 1988 . . . . . . . . . 10 wff 𝑤𝑦
1411, 1weq 1871 . . . . . . . . . 10 wff 𝑤 = 𝑦
1513, 14wo 383 . . . . . . . . 9 wff (𝑤𝑦𝑤 = 𝑦)
1612, 15wb 196 . . . . . . . 8 wff (𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))
1716, 11wal 1478 . . . . . . 7 wff 𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))
1810, 17wa 384 . . . . . 6 wff (𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
1918, 4wex 1701 . . . . 5 wff 𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
203, 19wi 4 . . . 4 wff (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120, 1wal 1478 . . 3 wff 𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
229, 21wa 384 . 2 wff (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2322, 2wex 1701 1 wff 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
Colors of variables: wff setvar class
This axiom is referenced by:  zfinf2  8483
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