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Theorem fin2i 8588
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 4549 . . . . 5  |-  ( A  e. FinII  ->  ~P A  e. 
_V )
2 elpw2g 4528 . . . . 5  |-  ( ~P A  e.  _V  ->  ( B  e.  ~P ~P A 
<->  B  C_  ~P A
) )
31, 2syl 16 . . . 4  |-  ( A  e. FinII  ->  ( B  e. 
~P ~P A  <->  B  C_  ~P A ) )
43biimpar 483 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  B  e.  ~P ~P A )
5 isfin2 8587 . . . . 5  |-  ( A  e. FinII  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
65ibi 241 . . . 4  |-  ( A  e. FinII  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) )
76adantr 463 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) )
8 neeq1 2663 . . . . . 6  |-  ( y  =  B  ->  (
y  =/=  (/)  <->  B  =/=  (/) ) )
9 soeq2 4734 . . . . . 6  |-  ( y  =  B  ->  ( [ C.] 
Or  y  <-> [ C.]  Or  B
) )
108, 9anbi12d 708 . . . . 5  |-  ( y  =  B  ->  (
( y  =/=  (/)  /\ [ C.]  Or  y
)  <->  ( B  =/=  (/)  /\ [ C.]  Or  B
) ) )
11 unieq 4171 . . . . . 6  |-  ( y  =  B  ->  U. y  =  U. B )
12 id 22 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
1311, 12eleq12d 2464 . . . . 5  |-  ( y  =  B  ->  ( U. y  e.  y  <->  U. B  e.  B ) )
1410, 13imbi12d 318 . . . 4  |-  ( y  =  B  ->  (
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  ( ( B  =/=  (/)  /\ [ C.]  Or  B
)  ->  U. B  e.  B ) ) )
1514rspcv 3131 . . 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 60 . 2  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  (
( B  =/=  (/)  /\ [ C.]  Or  B
)  ->  U. B  e.  B ) )
1716imp 427 1  |-  ( ( ( A  e. FinII  /\  B  C_ 
~P A )  /\  ( B  =/=  (/)  /\ [ C.]  Or  B
) )  ->  U. B  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    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-pow 4543
This theorem depends on definitions:  df-bi 185  df-an 369  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-in 3396  df-ss 3403  df-pw 3929  df-uni 4164  df-po 4714  df-so 4715  df-fin2 8579
This theorem is referenced by:  fin2i2  8611  ssfin2  8613  enfin2i  8614  fin1a2lem13  8705
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