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Theorem ssfin2 8200
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 3807 . . . . . 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 4413 . . . . . 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 3358 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P A )
8 fin2i 8175 . . . . 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 2789 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
12 ssexg 4349 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinII )  ->  B  e.  _V )
1312ancoms 440 . . 3  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e.  _V )
14 isfin2 8174 . . 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 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   U.cuni 4015    Or wor 4502   [ C.] crpss 6521  FinIIcfin2 8159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016  df-po 4503  df-so 4504  df-fin2 8166
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