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Theorem isfin4 10191
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 4046 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
2 breq2 5107 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
31, 2anbi12d 631 . . . 4 (𝑥 = 𝐴 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝐴𝑦𝐴)))
43exbidv 1924 . . 3 (𝑥 = 𝐴 → (∃𝑦(𝑦𝑥𝑦𝑥) ↔ ∃𝑦(𝑦𝐴𝑦𝐴)))
54notbid 317 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦𝑥𝑦𝑥) ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
6 df-fin4 10181 . 2 FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
75, 6elab2g 3630 1 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wpss 3909   class class class wbr 5103  cen 8838  FinIVcfin4 10174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-fin4 10181
This theorem is referenced by:  fin4i  10192  fin4en1  10203  ssfin4  10204  infpssALT  10207  isfin4-2  10208
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