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Theorem isfin7 8616
Description: Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin7  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
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 4387 . . . 4  |-  ( x  =  A  ->  (
x  ~~  y  <->  A  ~~  y ) )
21rexbidv 2910 . . 3  |-  ( x  =  A  ->  ( E. y  e.  ( On  \  om ) x 
~~  y  <->  E. y  e.  ( On  \  om ) A  ~~  y ) )
32notbid 292 . 2  |-  ( x  =  A  ->  ( -.  E. y  e.  ( On  \  om )
x  ~~  y  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
4 df-fin7 8606 . 2  |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
53, 4elab2g 3190 1  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1836   E.wrex 2747    \ cdif 3403   class class class wbr 4384   Oncon0 4809   omcom 6621    ~~ cen 7454  FinVIIcfin7 8599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-fin7 8606
This theorem is referenced by:  fin17  8709  fin67  8710  isfin7-2  8711
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