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Theorem isfin7 8573
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 4395 . . . 4  |-  ( x  =  A  ->  (
x  ~~  y  <->  A  ~~  y ) )
21rexbidv 2848 . . 3  |-  ( x  =  A  ->  ( E. y  e.  ( On  \  om ) x 
~~  y  <->  E. y  e.  ( On  \  om ) A  ~~  y ) )
32notbid 294 . 2  |-  ( x  =  A  ->  ( -.  E. y  e.  ( On  \  om )
x  ~~  y  <->  -.  E. y  e.  ( On  \  om ) A  ~~  y ) )
4 df-fin7 8563 . 2  |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
53, 4elab2g 3207 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 1370    e. wcel 1758   E.wrex 2796    \ cdif 3425   class class class wbr 4392   Oncon0 4819   omcom 6578    ~~ cen 7409  FinVIIcfin7 8556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-fin7 8563
This theorem is referenced by:  fin17  8666  fin67  8667  isfin7-2  8668
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