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Theorem ssfin2 8613
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 751 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  A  e. FinII )
2 elpwi 3936 . . . . . 6  |-  ( x  e.  ~P ~P B  ->  x  C_  ~P B
)
32adantl 464 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P B )
4 simplr 753 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  B  C_  A
)
5 sspwb 4611 . . . . . 6  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
64, 5sylib 196 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ~P B  C_  ~P A )
73, 6sstrd 3427 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P A )
8 fin2i 8588 . . . . 5  |-  ( ( ( A  e. FinII  /\  x  C_ 
~P A )  /\  ( x  =/=  (/)  /\ [ C.]  Or  x
) )  ->  U. x  e.  x )
98ex 432 . . . 4  |-  ( ( A  e. FinII  /\  x  C_  ~P A )  ->  (
( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
101, 7, 9syl2anc 659 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
1110ralrimiva 2796 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
12 ssexg 4511 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinII )  ->  B  e.  _V )
1312ancoms 451 . . 3  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e.  _V )
14 isfin2 8587 . . 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 232 1  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e. FinII )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1826    =/= wne 2577   A.wral 2732   _Vcvv 3034    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   U.cuni 4163    Or wor 4713   [ C.] crpss 6478  FinIIcfin2 8572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-pw 3929  df-sn 3945  df-pr 3947  df-uni 4164  df-po 4714  df-so 4715  df-fin2 8579
This theorem is referenced by: (None)
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