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Theorem fin2i 8664
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 4624 . . . . 5  |-  ( A  e. FinII  ->  ~P A  e. 
_V )
2 elpw2g 4603 . . . . 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 485 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  B  e.  ~P ~P A )
5 isfin2 8663 . . . . 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 465 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) )
8 neeq1 2741 . . . . . 6  |-  ( y  =  B  ->  (
y  =/=  (/)  <->  B  =/=  (/) ) )
9 soeq2 4813 . . . . . 6  |-  ( y  =  B  ->  ( [ C.]  Or  y  <-> [ C.]  Or  B
) )
108, 9anbi12d 710 . . . . 5  |-  ( y  =  B  ->  (
( y  =/=  (/)  /\ [ C.]  Or  y )  <->  ( B  =/=  (/)  /\ [ C.]  Or  B
) ) )
11 unieq 4246 . . . . . 6  |-  ( y  =  B  ->  U. y  =  U. B )
12 id 22 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
1311, 12eleq12d 2542 . . . . 5  |-  ( y  =  B  ->  ( U. y  e.  y  <->  U. B  e.  B ) )
1410, 13imbi12d 320 . . . 4  |-  ( y  =  B  ->  (
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  ( ( B  =/=  (/)  /\ [ C.]  Or  B
)  ->  U. B  e.  B ) ) )
1514rspcv 3203 . . 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 429 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 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   _Vcvv 3106    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238    Or wor 4792   [ C.] crpss 6554  FinIIcfin2 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-pow 4618
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-v 3108  df-in 3476  df-ss 3483  df-pw 4005  df-uni 4239  df-po 4793  df-so 4794  df-fin2 8655
This theorem is referenced by:  fin2i2  8687  ssfin2  8689  enfin2i  8690  fin1a2lem13  8781
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