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Theorem fin11a 8752
Description: Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin11a  |-  ( A  e.  Fin  ->  A  e. FinIa
)

Proof of Theorem fin11a
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elpwi 4012 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
2 ssfi 7730 . . . . 5  |-  ( ( A  e.  Fin  /\  x  C_  A )  ->  x  e.  Fin )
31, 2sylan2 474 . . . 4  |-  ( ( A  e.  Fin  /\  x  e.  ~P A
)  ->  x  e.  Fin )
43orcd 392 . . 3  |-  ( ( A  e.  Fin  /\  x  e.  ~P A
)  ->  ( x  e.  Fin  \/  ( A 
\  x )  e. 
Fin ) )
54ralrimiva 2871 . 2  |-  ( A  e.  Fin  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 isfin1a 8661 . 2  |-  ( A  e.  Fin  ->  ( A  e. FinIa 
<-> 
A. x  e.  ~P  A ( x  e. 
Fin  \/  ( A  \  x )  e.  Fin ) ) )
75, 6mpbird 232 1  |-  ( A  e.  Fin  ->  A  e. FinIa
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    e. wcel 1762   A.wral 2807    \ cdif 3466    C_ wss 3469   ~Pcpw 4003   Fincfn 7506  FinIacfin1a 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-om 6672  df-er 7301  df-en 7507  df-fin 7510  df-fin1a 8654
This theorem is referenced by: (None)
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