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Theorem fin2i 8206
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 4412 . . . . 5  |-  ( A  e. FinII  ->  ~P A  e. 
_V )
2 elpw2g 4392 . . . . 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 473 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  B  e.  ~P ~P A )
5 isfin2 8205 . . . . 5  |-  ( A  e. FinII  ->  ( A  e. FinII  <->  A. y  e.  ~P  ~P A
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
65ibi 234 . . . 4  |-  ( A  e. FinII  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) )
76adantr 453 . . 3  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) )
8 neeq1 2615 . . . . . 6  |-  ( y  =  B  ->  (
y  =/=  (/)  <->  B  =/=  (/) ) )
9 soeq2 4552 . . . . . 6  |-  ( y  =  B  ->  ( [ C.]  Or  y  <-> [ C.]  Or  B
) )
108, 9anbi12d 693 . . . . 5  |-  ( y  =  B  ->  (
( y  =/=  (/)  /\ [ C.]  Or  y )  <->  ( B  =/=  (/)  /\ [ C.]  Or  B
) ) )
11 unieq 4048 . . . . . 6  |-  ( y  =  B  ->  U. y  =  U. B )
12 id 21 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
1311, 12eleq12d 2510 . . . . 5  |-  ( y  =  B  ->  ( U. y  e.  y  <->  U. B  e.  B ) )
1410, 13imbi12d 313 . . . 4  |-  ( y  =  B  ->  (
( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  ( ( B  =/=  (/)  /\ [ C.]  Or  B
)  ->  U. B  e.  B ) ) )
1514rspcv 3054 . . 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 59 . 2  |-  ( ( A  e. FinII  /\  B  C_  ~P A )  ->  (
( B  =/=  (/)  /\ [ C.]  Or  B )  ->  U. B  e.  B ) )
1716imp 420 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 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   _Vcvv 2962    C_ wss 3306   (/)c0 3613   ~Pcpw 3823   U.cuni 4039    Or wor 4531   [ C.] crpss 6550  FinIIcfin2 8190
This theorem is referenced by:  fin2i2  8229  ssfin2  8231  enfin2i  8232  fin1a2lem13  8323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-pow 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-v 2964  df-in 3313  df-ss 3320  df-pw 3825  df-uni 4040  df-po 4532  df-so 4533  df-fin2 8197
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