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Theorem ssfin2 7946
Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
ssfin2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e. FinII )

Proof of Theorem ssfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 730 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  A  e. FinII )
2 elpwi 3633 . . . . . 6  |-  ( x  e.  ~P ~P B  ->  x  C_  ~P B
)
32adantl 452 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P B )
4 simplr 731 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  B  C_  A
)
5 sspwb 4223 . . . . . 6  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
64, 5sylib 188 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ~P B  C_  ~P A )
73, 6sstrd 3189 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P A )
8 fin2i 7921 . . . . 5  |-  ( ( ( A  e. FinII  /\  x  C_ 
~P A )  /\  ( x  =/=  (/)  /\ [ C.]  Or  x ) )  ->  U. x  e.  x
)
98ex 423 . . . 4  |-  ( ( A  e. FinII  /\  x  C_  ~P A )  ->  (
( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
101, 7, 9syl2anc 642 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
1110ralrimiva 2626 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
12 ssexg 4160 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinII )  ->  B  e.  _V )
1312ancoms 439 . . 3  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e.  _V )
14 isfin2 7920 . . 3  |-  ( B  e.  _V  ->  ( B  e. FinII 
<-> 
A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) ) )
1513, 14syl 15 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  ( B  e. FinII  <->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) ) )
1611, 15mpbird 223 1  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e. FinII )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827    Or wor 4313   [ C.] crpss 6276  FinIIcfin2 7905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-po 4314  df-so 4315  df-fin2 7912
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