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Theorem isfin1a 7934
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 3641 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 difeq1 3300 . . . . 5  |-  ( x  =  A  ->  (
x  \  y )  =  ( A  \ 
y ) )
32eleq1d 2362 . . . 4  |-  ( x  =  A  ->  (
( x  \  y
)  e.  Fin  <->  ( A  \  y )  e.  Fin ) )
43orbi2d 682 . . 3  |-  ( x  =  A  ->  (
( y  e.  Fin  \/  ( x  \  y
)  e.  Fin )  <->  ( y  e.  Fin  \/  ( A  \  y
)  e.  Fin )
) )
51, 4raleqbidv 2761 . 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 7927 . 2  |- FinIa  =  {
x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  (
x  \  y )  e.  Fin ) }
75, 6elab2g 2929 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 176    \/ wo 357    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162   ~Pcpw 3638   Fincfn 6879  FinIacfin1a 7920
This theorem is referenced by:  fin1ai  7935  fin11a  8025  enfin1ai  8026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-pw 3640  df-fin1a 7927
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