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Theorem isfin7 7927
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 4026 . . . 4  |-  ( x  =  A  ->  (
x  ~~  y  <->  A  ~~  y ) )
21rexbidv 2564 . . 3  |-  ( x  =  A  ->  ( E. y  e.  ( On  \  om ) x 
~~  y  <->  E. y  e.  ( On  \  om ) A  ~~  y ) )
32notbid 285 . 2  |-  ( x  =  A  ->  ( -.  E. y  e.  ( On  \  om )
x  ~~  y  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
4 df-fin7 7917 . 2  |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
53, 4elab2g 2916 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 176    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   class class class wbr 4023   Oncon0 4392   omcom 4656    ~~ cen 6860  FinVIIcfin7 7910
This theorem is referenced by:  fin17  8020  fin67  8021  isfin7-2  8022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-fin7 7917
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