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Theorem zfinf2 8652
Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8651 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
Assertion
Ref Expression
zfinf2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem zfinf2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 8651 . 2 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2 0el 4047 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
3 df-rex 3020 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
42, 3bitri 264 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
5 sucel 5911 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
6 df-rex 3020 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
75, 6bitri 264 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
87ralbii 3082 . . . . 5 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦𝑥𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
9 df-ral 3019 . . . . 5 (∀𝑦𝑥𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
108, 9bitri 264 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
114, 10anbi12i 735 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
1211exbii 1887 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ ∃𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
131, 12mpbir 221 1 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1594  wex 1817  wcel 2103  wral 3014  wrex 3015  c0 4023  suc csuc 5838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-inf2 8651
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-un 3685  df-nul 4024  df-sn 4286  df-suc 5842
This theorem is referenced by:  omex  8653
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