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Theorem fin1ai 7964
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 4211 . . 3  |-  ( A  e. FinIa  ->  ( X  e. 
~P A  <->  X  C_  A
) )
21biimpar 471 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  X  e.  ~P A )
3 isfin1a 7963 . . . 4  |-  ( A  e. FinIa  ->  ( A  e. FinIa  <->  A. x  e.  ~P  A
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )
) )
43ibi 232 . . 3  |-  ( A  e. FinIa  ->  A. x  e.  ~P  A ( x  e. 
Fin  \/  ( A  \  x )  e.  Fin ) )
54adantr 451 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 eleq1 2376 . . . 4  |-  ( x  =  X  ->  (
x  e.  Fin  <->  X  e.  Fin ) )
7 difeq2 3322 . . . . 5  |-  ( x  =  X  ->  ( A  \  x )  =  ( A  \  X
) )
87eleq1d 2382 . . . 4  |-  ( x  =  X  ->  (
( A  \  x
)  e.  Fin  <->  ( A  \  X )  e.  Fin ) )
96, 8orbi12d 690 . . 3  |-  ( x  =  X  ->  (
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )  <->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) ) )
109rspcv 2914 . 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 56 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 357    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    \ cdif 3183    C_ wss 3186   ~Pcpw 3659   Fincfn 6906  FinIacfin1a 7949
This theorem is referenced by:  enfin1ai  8055  fin1a2  8086  fin1aufil  17679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rab 2586  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-pw 3661  df-fin1a 7956
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