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Theorem ssfin2 8133
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 731 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  A  e. FinII )
2 elpwi 3750 . . . . . 6  |-  ( x  e.  ~P ~P B  ->  x  C_  ~P B
)
32adantl 453 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P B )
4 simplr 732 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  B  C_  A
)
5 sspwb 4354 . . . . . 6  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
64, 5sylib 189 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ~P B  C_  ~P A )
73, 6sstrd 3301 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P A )
8 fin2i 8108 . . . . 5  |-  ( ( ( A  e. FinII  /\  x  C_ 
~P A )  /\  ( x  =/=  (/)  /\ [ C.]  Or  x ) )  ->  U. x  e.  x
)
98ex 424 . . . 4  |-  ( ( A  e. FinII  /\  x  C_  ~P A )  ->  (
( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
101, 7, 9syl2anc 643 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
1110ralrimiva 2732 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
12 ssexg 4290 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinII )  ->  B  e.  _V )
1312ancoms 440 . . 3  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e.  _V )
14 isfin2 8107 . . 3  |-  ( B  e.  _V  ->  ( B  e. FinII 
<-> 
A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) ) )
1513, 14syl 16 . 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 224 1  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e. FinII )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957    Or wor 4443   [ C.] crpss 6457  FinIIcfin2 8092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-pw 3744  df-sn 3763  df-pr 3764  df-uni 3958  df-po 4444  df-so 4445  df-fin2 8099
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