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Theorem fin12 8601
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8603. (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 2994 . . . . . . . 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 8574 . . . . . . . . 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 3888 . . . . . . . 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 4701 . . . . . . 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 1219 . . . . . 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 2994 . . . . . . . . . . 11  |-  d  e. 
_V
12 vex 2994 . . . . . . . . . . 11  |-  c  e. 
_V
1311, 12brcnv 5041 . . . . . . . . . 10  |-  ( d `' [ C.]  c  <->  c [ C.]  d )
1411brrpss 6382 . . . . . . . . . 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 2758 . . . . . . 7  |-  ( A. d  e.  b  -.  d `' [ C.]  c  <->  A. d  e.  b  -.  c  C.  d )
1817rexbii 2759 . . . . . 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 6388 . . . . . 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 2818 . 2  |-  ( A  e.  Fin  ->  A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
25 isfin2 8482 . 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 1756    =/= wne 2620   A.wral 2734   E.wrex 2735   _Vcvv 2991    C_ wss 3347    C. wpss 3348   (/)c0 3656   ~Pcpw 3879   U.cuni 4110   class class class wbr 4311    Or wor 4659    Fr wfr 4695   `'ccnv 4858   [ C.] crpss 6378   Fincfn 7329  FinIIcfin2 8467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-rpss 6379  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fin2 8474
This theorem is referenced by:  fin1a2s  8602  fin1a2  8603  finngch  8841
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