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Theorem isfin4 8979
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 3656 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
2 breq2 4581 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
31, 2anbi12d 742 . . . 4 (𝑥 = 𝐴 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝐴𝑦𝐴)))
43exbidv 1836 . . 3 (𝑥 = 𝐴 → (∃𝑦(𝑦𝑥𝑦𝑥) ↔ ∃𝑦(𝑦𝐴𝑦𝐴)))
54notbid 306 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦𝑥𝑦𝑥) ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
6 df-fin4 8969 . 2 FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
75, 6elab2g 3321 1 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wpss 3540   class class class wbr 4577  cen 7815  FinIVcfin4 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-fin4 8969
This theorem is referenced by:  fin4i  8980  fin4en1  8991  ssfin4  8992  infpssALT  8995  isfin4-2  8996
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