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Theorem isfin1a 8624
Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin1a  |-  ( A  e.  V  ->  ( A  e. FinIa 
<-> 
A. y  e.  ~P  A ( y  e. 
Fin  \/  ( A  \  y )  e.  Fin ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin1a
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3957 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 difeq1 3553 . . . . 5  |-  ( x  =  A  ->  (
x  \  y )  =  ( A  \ 
y ) )
32eleq1d 2471 . . . 4  |-  ( x  =  A  ->  (
( x  \  y
)  e.  Fin  <->  ( A  \  y )  e.  Fin ) )
43orbi2d 700 . . 3  |-  ( x  =  A  ->  (
( y  e.  Fin  \/  ( x  \  y
)  e.  Fin )  <->  ( y  e.  Fin  \/  ( A  \  y
)  e.  Fin )
) )
51, 4raleqbidv 3017 . 2  |-  ( x  =  A  ->  ( A. y  e.  ~P  x ( y  e. 
Fin  \/  ( x  \  y )  e.  Fin ) 
<-> 
A. y  e.  ~P  A ( y  e. 
Fin  \/  ( A  \  y )  e.  Fin ) ) )
6 df-fin1a 8617 . 2  |- FinIa  =  {
x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  (
x  \  y )  e.  Fin ) }
75, 6elab2g 3197 1  |-  ( A  e.  V  ->  ( A  e. FinIa 
<-> 
A. y  e.  ~P  A ( y  e. 
Fin  \/  ( A  \  y )  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    = wceq 1405    e. wcel 1842   A.wral 2753    \ cdif 3410   ~Pcpw 3954   Fincfn 7474  FinIacfin1a 8610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-in 3420  df-ss 3427  df-pw 3956  df-fin1a 8617
This theorem is referenced by:  fin1ai  8625  fin11a  8715  enfin1ai  8716
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