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Theorem fin2i 8568
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 4577 . . . . 5  |-  ( A  e. FinII  ->  ~P A  e. 
_V )
2 elpw2g 4556 . . . . 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 8567 . . . . 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 2729 . . . . . 6  |-  ( y  =  B  ->  (
y  =/=  (/)  <->  B  =/=  (/) ) )
9 soeq2 4762 . . . . . 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 4200 . . . . . 6  |-  ( y  =  B  ->  U. y  =  U. B )
12 id 22 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
1311, 12eleq12d 2533 . . . . 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 3168 . . 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 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   _Vcvv 3071    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   U.cuni 4192    Or wor 4741   [ C.] crpss 6462  FinIIcfin2 8552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-pow 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-pw 3963  df-uni 4193  df-po 4742  df-so 4743  df-fin2 8559
This theorem is referenced by:  fin2i2  8591  ssfin2  8593  enfin2i  8594  fin1a2lem13  8685
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