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Theorem fin2i 7937
Description: Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin2i  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  U. B  e.  B
)

Proof of Theorem fin2i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pwexg 4210 . . . . 5  |-  ( A  e. FinII  ->  ~P A  e. 
_V )
2 elpw2g 4190 . . . . 5  |-  ( ~P A  e.  _V  ->  ( B  e.  ~P ~P A 
<->  B  C_  ~P A
) )
31, 2syl 15 . . . 4  |-  ( A  e. FinII  ->  ( B  e. 
~P ~P A  <->  B  C_  ~P A ) )
43biimpar 471 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  B  e.  ~P ~P A )
5 isfin2 7936 . . . . 5  |-  ( A  e. FinII  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
65ibi 232 . . . 4  |-  ( A  e. FinII  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) )
76adantr 451 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) )
8 neeq1 2467 . . . . . 6  |-  ( y  =  B  ->  (
y  =/=  (/)  <->  B  =/=  (/) ) )
9 soeq2 4350 . . . . . 6  |-  ( y  =  B  ->  ( [ C.]  Or  y  <-> [ C.]  Or  B
) )
108, 9anbi12d 691 . . . . 5  |-  ( y  =  B  ->  (
( y  =/=  (/)  /\ [ C.]  Or  y )  <->  ( B  =/=  (/)  /\ [ C.]  Or  B
) ) )
11 unieq 3852 . . . . . 6  |-  ( y  =  B  ->  U. y  =  U. B )
12 id 19 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
1311, 12eleq12d 2364 . . . . 5  |-  ( y  =  B  ->  ( U. y  e.  y  <->  U. B  e.  B ) )
1410, 13imbi12d 311 . . . 4  |-  ( y  =  B  ->  (
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  ( ( B  =/=  (/)  /\ [ C.]  Or  B
)  ->  U. B  e.  B ) ) )
1514rspcv 2893 . . 3  |-  ( B  e.  ~P ~P A  ->  ( A. y  e. 
~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y )  ->  (
( B  =/=  (/)  /\ [ C.]  Or  B )  ->  U. B  e.  B ) ) )
164, 7, 15sylc 56 . 2  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  (
( B  =/=  (/)  /\ [ C.]  Or  B )  ->  U. B  e.  B ) )
1716imp 418 1  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B ) )  ->  U. B  e.  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843    Or wor 4329   [ C.] crpss 6292  FinIIcfin2 7921
This theorem is referenced by:  fin2i2  7960  ssfin2  7962  enfin2i  7963  fin1a2lem13  8054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-po 4330  df-so 4331  df-fin2 7928
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