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Theorem isfin7 9406
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 4845 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
21rexbidv 3238 . . 3 (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
32notbid 309 . 2 (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
4 df-fin7 9396 . 2 FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
53, 4elab2g 3546 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1637  wcel 2156  wrex 3095  cdif 3764   class class class wbr 4842  Oncon0 5934  ωcom 7293  cen 8187  FinVIIcfin7 9389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-rex 3100  df-rab 3103  df-v 3391  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4843  df-fin7 9396
This theorem is referenced by:  fin17  9499  fin67  9500  isfin7-2  9501
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