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Theorem isfin4 10335
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 4101 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
2 breq2 5152 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
31, 2anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑦𝑥𝑦𝑥) ↔ (𝑦𝐴𝑦𝐴)))
43exbidv 1919 . . 3 (𝑥 = 𝐴 → (∃𝑦(𝑦𝑥𝑦𝑥) ↔ ∃𝑦(𝑦𝐴𝑦𝐴)))
54notbid 318 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦𝑥𝑦𝑥) ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
6 df-fin4 10325 . 2 FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
75, 6elab2g 3683 1 (𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wpss 3964   class class class wbr 5148  cen 8981  FinIVcfin4 10318
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fin4 10325
This theorem is referenced by:  fin4i  10336  fin4en1  10347  ssfin4  10348  infpssALT  10351  isfin4-2  10352
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