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Theorem isfin1a 8177
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 3804 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 difeq1 3460 . . . . 5  |-  ( x  =  A  ->  (
x  \  y )  =  ( A  \ 
y ) )
32eleq1d 2504 . . . 4  |-  ( x  =  A  ->  (
( x  \  y
)  e.  Fin  <->  ( A  \  y )  e.  Fin ) )
43orbi2d 684 . . 3  |-  ( x  =  A  ->  (
( y  e.  Fin  \/  ( x  \  y
)  e.  Fin )  <->  ( y  e.  Fin  \/  ( A  \  y
)  e.  Fin )
) )
51, 4raleqbidv 2918 . 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 8170 . 2  |- FinIa  =  {
x  |  A. y  e.  ~P  x ( y  e.  Fin  \/  (
x  \  y )  e.  Fin ) }
75, 6elab2g 3086 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 178    \/ wo 359    = wceq 1653    e. wcel 1726   A.wral 2707    \ cdif 3319   ~Pcpw 3801   Fincfn 7112  FinIacfin1a 8163
This theorem is referenced by:  fin1ai  8178  fin11a  8268  enfin1ai  8269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-pw 3803  df-fin1a 8170
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