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Theorem fin12 8805
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8807. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin12  |-  ( A  e.  Fin  ->  A  e. FinII
)

Proof of Theorem fin12
Dummy variables  b 
c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3121 . . . . . . . 8  |-  b  e. 
_V
21a1i 11 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  b  e.  _V )
3 isfin1-3 8778 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  e.  Fin  <->  `' [ C.]  Fr  ~P A ) )
43ibi 241 . . . . . . . 8  |-  ( A  e.  Fin  ->  `' [ C.] 
Fr  ~P A )
54ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  `' [ C.] 
Fr  ~P A )
6 elpwi 4025 . . . . . . . 8  |-  ( b  e.  ~P ~P A  ->  b  C_  ~P A
)
76ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  b  C_ 
~P A )
8 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  b  =/=  (/) )
9 fri 4847 . . . . . . 7  |-  ( ( ( b  e.  _V  /\  `' [ C.]  Fr  ~P A
)  /\  ( b  C_ 
~P A  /\  b  =/=  (/) ) )  ->  E. c  e.  b  A. d  e.  b  -.  d `' [ C.]  c
)
102, 5, 7, 8, 9syl22anc 1229 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  E. c  e.  b  A. d  e.  b  -.  d `' [ C.]  c )
11 vex 3121 . . . . . . . . . . 11  |-  d  e. 
_V
12 vex 3121 . . . . . . . . . . 11  |-  c  e. 
_V
1311, 12brcnv 5191 . . . . . . . . . 10  |-  ( d `' [ C.]  c  <->  c [ C.]  d
)
1411brrpss 6578 . . . . . . . . . 10  |-  ( c [ C.]  d  <->  c  C.  d
)
1513, 14bitri 249 . . . . . . . . 9  |-  ( d `' [ C.]  c  <->  c  C.  d
)
1615notbii 296 . . . . . . . 8  |-  ( -.  d `' [ C.]  c  <->  -.  c  C.  d )
1716ralbii 2898 . . . . . . 7  |-  ( A. d  e.  b  -.  d `' [ C.]  c  <->  A. d  e.  b  -.  c  C.  d )
1817rexbii 2969 . . . . . 6  |-  ( E. c  e.  b  A. d  e.  b  -.  d `' [ C.]  c  <->  E. c  e.  b  A. d  e.  b  -.  c  C.  d )
1910, 18sylib 196 . . . . 5  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  E. c  e.  b  A. d  e.  b  -.  c  C.  d )
20 sorpssuni 6584 . . . . . 6  |-  ( [ C.]  Or  b  ->  ( E. c  e.  b  A. d  e.  b  -.  c  C.  d  <->  U. b  e.  b ) )
2120ad2antll 728 . . . . 5  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  ( E. c  e.  b  A. d  e.  b  -.  c  C.  d  <->  U. b  e.  b ) )
2219, 21mpbid 210 . . . 4  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b
) )  ->  U. b  e.  b )
2322ex 434 . . 3  |-  ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  ->  (
( b  =/=  (/)  /\ [ C.]  Or  b
)  ->  U. b  e.  b ) )
2423ralrimiva 2881 . 2  |-  ( A  e.  Fin  ->  A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b
)  ->  U. b  e.  b ) )
25 isfin2 8686 . 2  |-  ( A  e.  Fin  ->  ( A  e. FinII 
<-> 
A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b
)  ->  U. b  e.  b ) ) )
2624, 25mpbird 232 1  |-  ( A  e.  Fin  ->  A  e. FinII
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481    C. wpss 3482   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   class class class wbr 4453    Or wor 4805    Fr wfr 4841   `'ccnv 5004   [ C.] crpss 6574   Fincfn 7528  FinIIcfin2 8671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rpss 6575  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fin2 8678
This theorem is referenced by:  fin1a2s  8806  fin1a2  8807  finngch  9045
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