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Theorem fin11a 8656
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 3970 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
2 ssfi 7637 . . . . 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 2825 . 2  |-  ( A  e.  Fin  ->  A. x  e.  ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. 
Fin ) )
6 isfin1a 8565 . 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 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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-om 6580  df-er 7204  df-en 7414  df-fin 7417  df-fin1a 8558
This theorem is referenced by: (None)
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