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Theorem zfinf2 9680
Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9679 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 9679 . 2 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2 0el 4369 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
3 df-rex 3069 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
42, 3bitri 275 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
5 sucel 6460 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
6 df-rex 3069 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
75, 6bitri 275 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
87ralbii 3091 . . . . 5 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦𝑥𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
9 df-ral 3060 . . . . 5 (∀𝑦𝑥𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
108, 9bitri 275 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
114, 10anbi12i 628 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
1211exbii 1845 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ ∃𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
131, 12mpbir 231 1 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  wal 1535  wex 1776  wcel 2106  wral 3059  wrex 3068  c0 4339  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-suc 6392
This theorem is referenced by:  omex  9681
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