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Theorem isfin7 9083
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 4626 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
21rexbidv 3047 . . 3 (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
32notbid 308 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
4 df-fin7 9073 . 2 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
53, 4elab2g 3341 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wcel 1987  wrex 2909  cdif 3557   class class class wbr 4623  Oncon0 5692  ωcom 7027  cen 7912  FinVIIcfin7 9066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-fin7 9073
This theorem is referenced by:  fin17  9176  fin67  9177  isfin7-2  9178
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