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Theorem isfin7 10339
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 5151 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
21rexbidv 3177 . . 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 10329 . 2 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
53, 4elab2g 3683 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wrex 3068  cdif 3960   class class class wbr 5148  Oncon0 6386  ωcom 7887  cen 8981  FinVIIcfin7 10322
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-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fin7 10329
This theorem is referenced by:  fin17  10432  fin67  10433  isfin7-2  10434
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