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Theorem fin12 8039
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8041. (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 2791 . . . . . . . 8  |-  b  e. 
_V
21a1i 10 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  -> 
b  e.  _V )
3 isfin1-3 8012 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  e.  Fin  <->  `' [ C.]  Fr  ~P A ) )
43ibi 232 . . . . . . . 8  |-  ( A  e.  Fin  ->  `' [ C.] 
Fr  ~P A )
54ad2antrr 706 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  ->  `' [ C.]  Fr  ~P A
)
6 elpwi 3633 . . . . . . . 8  |-  ( b  e.  ~P ~P A  ->  b  C_  ~P A
)
76ad2antlr 707 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  -> 
b  C_  ~P A
)
8 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  -> 
b  =/=  (/) )
9 fri 4355 . . . . . . 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 1183 . . . . . 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 2791 . . . . . . . . . . 11  |-  d  e. 
_V
12 vex 2791 . . . . . . . . . . 11  |-  c  e. 
_V
1311, 12brcnv 4864 . . . . . . . . . 10  |-  ( d `' [ C.]  c  <->  c [ C.]  d )
1411brrpss 6280 . . . . . . . . . 10  |-  ( c [
C.]  d  <->  c  C.  d )
1513, 14bitri 240 . . . . . . . . 9  |-  ( d `' [ C.]  c  <->  c  C.  d )
1615notbii 287 . . . . . . . 8  |-  ( -.  d `' [ C.]  c  <->  -.  c  C.  d )
1716ralbii 2567 . . . . . . 7  |-  ( A. d  e.  b  -.  d `' [ C.]  c  <->  A. d  e.  b  -.  c  C.  d )
1817rexbii 2568 . . . . . 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 188 . . . . 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 6286 . . . . . 6  |-  ( [ C.]  Or  b  ->  ( E. c  e.  b  A. d  e.  b  -.  c  C.  d  <->  U. b  e.  b ) )
2120ad2antll 709 . . . . 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 201 . . . 4  |-  ( ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  /\  (
b  =/=  (/)  /\ [ C.]  Or  b ) )  ->  U. b  e.  b
)
2322ex 423 . . 3  |-  ( ( A  e.  Fin  /\  b  e.  ~P ~P A )  ->  (
( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
2423ralrimiva 2626 . 2  |-  ( A  e.  Fin  ->  A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
25 isfin2 7920 . 2  |-  ( A  e.  Fin  ->  ( A  e. FinII 
<-> 
A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b
)  ->  U. b  e.  b ) ) )
2624, 25mpbird 223 1  |-  ( A  e.  Fin  ->  A  e. FinII
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152    C. wpss 3153   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023    Or wor 4313    Fr wfr 4349   `'ccnv 4688   [ C.] crpss 6276   Fincfn 6863  FinIIcfin2 7905
This theorem is referenced by:  fin1a2s  8040  fin1a2  8041  finngch  8277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rpss 6277  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fin2 7912
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