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Theorem isfin4 10338
Description: Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin4 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 psseq2 4090 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
2 breq2 5146 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
31, 2anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝐴𝑦𝐴)))
43exbidv 1920 . . 3 (𝑥 = 𝐴 → (∃𝑦(𝑦𝑥𝑦𝑥) ↔ ∃𝑦(𝑦𝐴𝑦𝐴)))
54notbid 318 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦𝑥𝑦𝑥) ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
6 df-fin4 10328 . 2 FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
75, 6elab2g 3679 1 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wpss 3951   class class class wbr 5142  cen 8983  FinIVcfin4 10321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-fin4 10328
This theorem is referenced by:  fin4i  10339  fin4en1  10350  ssfin4  10351  infpssALT  10354  isfin4-2  10355
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