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Theorem fin1ai 8173
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 4363 . . 3  |-  ( A  e. FinIa  ->  ( X  e. 
~P A  <->  X  C_  A
) )
21biimpar 472 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  X  e.  ~P A )
3 isfin1a 8172 . . . 4  |-  ( A  e. FinIa  ->  ( A  e. FinIa  <->  A. x  e.  ~P  A
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )
) )
43ibi 233 . . 3  |-  ( A  e. FinIa  ->  A. x  e.  ~P  A ( x  e. 
Fin  \/  ( A  \  x )  e.  Fin ) )
54adantr 452 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 eleq1 2496 . . . 4  |-  ( x  =  X  ->  (
x  e.  Fin  <->  X  e.  Fin ) )
7 difeq2 3459 . . . . 5  |-  ( x  =  X  ->  ( A  \  x )  =  ( A  \  X
) )
87eleq1d 2502 . . . 4  |-  ( x  =  X  ->  (
( A  \  x
)  e.  Fin  <->  ( A  \  X )  e.  Fin ) )
96, 8orbi12d 691 . . 3  |-  ( x  =  X  ->  (
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )  <->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) ) )
109rspcv 3048 . 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 58 1  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  ( X  e.  Fin  \/  ( A 
\  X )  e. 
Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    \ cdif 3317    C_ wss 3320   ~Pcpw 3799   Fincfn 7109  FinIacfin1a 8158
This theorem is referenced by:  enfin1ai  8264  fin1a2  8295  fin1aufil  17964
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 2417  ax-sep 4330
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-pw 3801  df-fin1a 8165
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