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Theorem isfin1a 8164
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 3794 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 difeq1 3450 . . . . 5  |-  ( x  =  A  ->  (
x  \  y )  =  ( A  \ 
y ) )
32eleq1d 2501 . . . 4  |-  ( x  =  A  ->  (
( x  \  y
)  e.  Fin  <->  ( A  \  y )  e.  Fin ) )
43orbi2d 683 . . 3  |-  ( x  =  A  ->  (
( y  e.  Fin  \/  ( x  \  y
)  e.  Fin )  <->  ( y  e.  Fin  \/  ( A  \  y
)  e.  Fin )
) )
51, 4raleqbidv 2908 . 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 8157 . 2  |- FinIa  =  {
x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  (
x  \  y )  e.  Fin ) }
75, 6elab2g 3076 1  |-  ( A  e.  V  ->  ( A  e. FinIa 
<-> 
A. y  e.  ~P  A ( y  e. 
Fin  \/  ( A  \  y )  e.  Fin ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725   A.wral 2697    \ cdif 3309   ~Pcpw 3791   Fincfn 7101  FinIacfin1a 8150
This theorem is referenced by:  fin1ai  8165  fin11a  8255  enfin1ai  8256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-pw 3793  df-fin1a 8157
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