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Theorem isfin4 9711
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 4068 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
2 breq2 5066 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
31, 2anbi12d 630 . . . 4 (𝑥 = 𝐴 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝐴𝑦𝐴)))
43exbidv 1915 . . 3 (𝑥 = 𝐴 → (∃𝑦(𝑦𝑥𝑦𝑥) ↔ ∃𝑦(𝑦𝐴𝑦𝐴)))
54notbid 319 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦𝑥𝑦𝑥) ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
6 df-fin4 9701 . 2 FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
75, 6elab2g 3672 1 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2106  wpss 3940   class class class wbr 5062  cen 8498  FinIVcfin4 9694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-fin4 9701
This theorem is referenced by:  fin4i  9712  fin4en1  9723  ssfin4  9724  infpssALT  9727  isfin4-2  9728
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