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Theorem isfin7 10057
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 5077 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
21rexbidv 3226 . . 3 (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
32notbid 318 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
4 df-fin7 10047 . 2 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
53, 4elab2g 3611 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1539  wcel 2106  wrex 3065  cdif 3884   class class class wbr 5074  Oncon0 6266  ωcom 7712  cen 8730  FinVIIcfin7 10040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-fin7 10047
This theorem is referenced by:  fin17  10150  fin67  10151  isfin7-2  10152
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