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Theorem isfin7 10195
Description: Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin7 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin7
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq1 5106 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
21rexbidv 3173 . . 3 (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
32notbid 317 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
4 df-fin7 10185 . 2 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
53, 4elab2g 3630 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  wrex 3071  cdif 3905   class class class wbr 5103  Oncon0 6315  ωcom 7794  cen 8838  FinVIIcfin7 10178
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-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-fin7 10185
This theorem is referenced by:  fin17  10288  fin67  10289  isfin7-2  10290
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