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Theorem isfin7 8182
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 4216 . . . 4  |-  ( x  =  A  ->  (
x  ~~  y  <->  A  ~~  y ) )
21rexbidv 2727 . . 3  |-  ( x  =  A  ->  ( E. y  e.  ( On  \  om ) x 
~~  y  <->  E. y  e.  ( On  \  om ) A  ~~  y ) )
32notbid 287 . 2  |-  ( x  =  A  ->  ( -.  E. y  e.  ( On  \  om )
x  ~~  y  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
4 df-fin7 8172 . 2  |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
53, 4elab2g 3085 1  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2707    \ cdif 3318   class class class wbr 4213   Oncon0 4582   omcom 4846    ~~ cen 7107  FinVIIcfin7 8165
This theorem is referenced by:  fin17  8275  fin67  8276  isfin7-2  8277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-fin7 8172
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