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Theorem fin1ai 8566
Description: Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin1ai  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  ( X  e.  Fin  \/  ( A 
\  X )  e. 
Fin ) )

Proof of Theorem fin1ai
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elpw2g 4556 . . 3  |-  ( A  e. FinIa  ->  ( X  e. 
~P A  <->  X  C_  A
) )
21biimpar 485 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  X  e.  ~P A )
3 isfin1a 8565 . . . 4  |-  ( A  e. FinIa  ->  ( A  e. FinIa  <->  A. x  e.  ~P  A
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )
) )
43ibi 241 . . 3  |-  ( A  e. FinIa  ->  A. x  e.  ~P  A ( x  e. 
Fin  \/  ( A  \  x )  e.  Fin ) )
54adantr 465 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 eleq1 2523 . . . 4  |-  ( x  =  X  ->  (
x  e.  Fin  <->  X  e.  Fin ) )
7 difeq2 3569 . . . . 5  |-  ( x  =  X  ->  ( A  \  x )  =  ( A  \  X
) )
87eleq1d 2520 . . . 4  |-  ( x  =  X  ->  (
( A  \  x
)  e.  Fin  <->  ( A  \  X )  e.  Fin ) )
96, 8orbi12d 709 . . 3  |-  ( x  =  X  ->  (
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )  <->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) ) )
109rspcv 3168 . 2  |-  ( X  e.  ~P A  -> 
( A. x  e. 
~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin )  ->  ( X  e.  Fin  \/  ( A  \  X )  e. 
Fin ) ) )
112, 5, 10sylc 60 1  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  ( X  e.  Fin  \/  ( A 
\  X )  e. 
Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    \ cdif 3426    C_ wss 3429   ~Pcpw 3961   Fincfn 7413  FinIacfin1a 8551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-pw 3963  df-fin1a 8558
This theorem is referenced by:  enfin1ai  8657  fin1a2  8688  fin1aufil  19630
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