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Theorem isfin7 8677
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 4450 . . . 4  |-  ( x  =  A  ->  (
x  ~~  y  <->  A  ~~  y ) )
21rexbidv 2973 . . 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 8667 . 2  |- FinVII  =  { x  |  -.  E. y  e.  ( On  \  om ) x  ~~  y }
53, 4elab2g 3252 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 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473   class class class wbr 4447   Oncon0 4878   omcom 6678    ~~ cen 7510  FinVIIcfin7 8660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-fin7 8667
This theorem is referenced by:  fin17  8770  fin67  8771  isfin7-2  8772
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