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Theorem ssfin2 8696
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 753 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  A  e. FinII )
2 elpwi 4019 . . . . . 6  |-  ( x  e.  ~P ~P B  ->  x  C_  ~P B
)
32adantl 466 . . . . 5  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P B )
4 simplr 754 . . . . . 6  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  B  C_  A
)
5 sspwb 4696 . . . . . 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 3514 . . . 4  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  x  C_  ~P A )
8 fin2i 8671 . . . . 5  |-  ( ( ( A  e. FinII  /\  x  C_ 
~P A )  /\  ( x  =/=  (/)  /\ [ C.]  Or  x ) )  ->  U. x  e.  x
)
98ex 434 . . . 4  |-  ( ( A  e. FinII  /\  x  C_  ~P A )  ->  (
( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
101, 7, 9syl2anc 661 . . 3  |-  ( ( ( A  e. FinII  /\  B  C_  A )  /\  x  e.  ~P ~P B )  ->  ( ( x  =/=  (/)  /\ [ C.]  Or  x
)  ->  U. x  e.  x ) )
1110ralrimiva 2878 . 2  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  A. x  e.  ~P  ~P B ( ( x  =/=  (/)  /\ [ C.]  Or  x )  ->  U. x  e.  x ) )
12 ssexg 4593 . . . 4  |-  ( ( B  C_  A  /\  A  e. FinII )  ->  B  e.  _V )
1312ancoms 453 . . 3  |-  ( ( A  e. FinII  /\  B  C_  A
)  ->  B  e.  _V )
14 isfin2 8670 . . 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 369    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245    Or wor 4799   [ C.] crpss 6561  FinIIcfin2 8655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-uni 4246  df-po 4800  df-so 4801  df-fin2 8662
This theorem is referenced by: (None)
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