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Theorem fin12 8295
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8297. (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 2961 . . . . . . . 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 8268 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  e.  Fin  <->  `' [ C.]  Fr  ~P A ) )
43ibi 234 . . . . . . . 8  |-  ( A  e.  Fin  ->  `' [ C.] 
Fr  ~P A )
54ad2antrr 708 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  ->  `' [ C.]  Fr  ~P A
)
6 elpwi 3809 . . . . . . . 8  |-  ( b  e.  ~P ~P A  ->  b  C_  ~P A
)
76ad2antlr 709 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  -> 
b  C_  ~P A
)
8 simprl 734 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  -> 
b  =/=  (/) )
9 fri 4546 . . . . . . 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 1186 . . . . . 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 2961 . . . . . . . . . . 11  |-  d  e. 
_V
12 vex 2961 . . . . . . . . . . 11  |-  c  e. 
_V
1311, 12brcnv 5057 . . . . . . . . . 10  |-  ( d `' [ C.]  c  <->  c [ C.]  d )
1411brrpss 6527 . . . . . . . . . 10  |-  ( c [
C.]  d  <->  c  C.  d )
1513, 14bitri 242 . . . . . . . . 9  |-  ( d `' [ C.]  c  <->  c  C.  d )
1615notbii 289 . . . . . . . 8  |-  ( -.  d `' [ C.]  c  <->  -.  c  C.  d )
1716ralbii 2731 . . . . . . 7  |-  ( A. d  e.  b  -.  d `' [ C.]  c  <->  A. d  e.  b  -.  c  C.  d )
1817rexbii 2732 . . . . . 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 190 . . . . 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 6533 . . . . . 6  |-  ( [ C.]  Or  b  ->  ( E. c  e.  b  A. d  e.  b  -.  c  C.  d  <->  U. b  e.  b ) )
2120ad2antll 711 . . . . 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 203 . . . 4  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  ->  U. b  e.  b
)
2322ex 425 . . 3  |-  ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  ->  (
( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
2423ralrimiva 2791 . 2  |-  ( A  e.  Fin  ->  A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
25 isfin2 8176 . 2  |-  ( A  e.  Fin  ->  ( A  e. FinII 
<-> 
A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b
)  ->  U. b  e.  b ) ) )
2624, 25mpbird 225 1  |-  ( A  e.  Fin  ->  A  e. FinII
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322    C. wpss 3323   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   class class class wbr 4214    Or wor 4504    Fr wfr 4540   `'ccnv 4879   [ C.] crpss 6523   Fincfn 7111  FinIIcfin2 8161
This theorem is referenced by:  fin1a2s  8296  fin1a2  8297  finngch  8532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-rpss 6524  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fin2 8168
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