MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1ai Structured version   Unicode version

Theorem fin1ai 8664
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 4600 . . 3  |-  ( A  e. FinIa  ->  ( X  e. 
~P A  <->  X  C_  A
) )
21biimpar 483 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  X  e.  ~P A )
3 isfin1a 8663 . . . 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 463 . 2  |-  ( ( A  e. FinIa  /\  X  C_  A
)  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 eleq1 2526 . . . 4  |-  ( x  =  X  ->  (
x  e.  Fin  <->  X  e.  Fin ) )
7 difeq2 3602 . . . . 5  |-  ( x  =  X  ->  ( A  \  x )  =  ( A  \  X
) )
87eleq1d 2523 . . . 4  |-  ( x  =  X  ->  (
( A  \  x
)  e.  Fin  <->  ( A  \  X )  e.  Fin ) )
96, 8orbi12d 707 . . 3  |-  ( x  =  X  ->  (
( x  e.  Fin  \/  ( A  \  x
)  e.  Fin )  <->  ( X  e.  Fin  \/  ( A  \  X )  e.  Fin ) ) )
109rspcv 3203 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    \ cdif 3458    C_ wss 3461   ~Pcpw 3999   Fincfn 7509  FinIacfin1a 8649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pw 4001  df-fin1a 8656
This theorem is referenced by:  enfin1ai  8755  fin1a2  8786  fin1aufil  20599
  Copyright terms: Public domain W3C validator