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Axiom ax-inf2 8699
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 8700 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8698 above, using our version of Infinity ax-inf 8696 and the Axiom of Regularity ax-reg 8650. We will reference ax-inf2 8699 instead of axinf2 8698 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8696 from ax-inf2 8699 is shown by theorem axinf 8702. (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 2128 . . . . 5 wff 𝑦𝑥
4 vz . . . . . . . 8 setvar 𝑧
54, 1wel 2128 . . . . . . 7 wff 𝑧𝑦
65wn 3 . . . . . 6 wff ¬ 𝑧𝑦
76, 4wal 1618 . . . . 5 wff 𝑧 ¬ 𝑧𝑦
83, 7wa 383 . . . 4 wff (𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦)
98, 1wex 1841 . . 3 wff 𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦)
104, 2wel 2128 . . . . . . 7 wff 𝑧𝑥
11 vw . . . . . . . . . 10 setvar 𝑤
1211, 4wel 2128 . . . . . . . . 9 wff 𝑤𝑧
1311, 1wel 2128 . . . . . . . . . 10 wff 𝑤𝑦
1411, 1weq 2028 . . . . . . . . . 10 wff 𝑤 = 𝑦
1513, 14wo 382 . . . . . . . . 9 wff (𝑤𝑦𝑤 = 𝑦)
1612, 15wb 196 . . . . . . . 8 wff (𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))
1716, 11wal 1618 . . . . . . 7 wff 𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))
1810, 17wa 383 . . . . . 6 wff (𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
1918, 4wex 1841 . . . . 5 wff 𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
203, 19wi 4 . . . 4 wff (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120, 1wal 1618 . . 3 wff 𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
229, 21wa 383 . 2 wff (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2322, 2wex 1841 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  8700
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