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Theorem fin12 8055
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8057. (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 2804 . . . . . . . 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 8028 . . . . . . . . 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 3646 . . . . . . . 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 4371 . . . . . . 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 2804 . . . . . . . . . . 11  |-  d  e. 
_V
12 vex 2804 . . . . . . . . . . 11  |-  c  e. 
_V
1311, 12brcnv 4880 . . . . . . . . . 10  |-  ( d `' [ C.]  c  <->  c [ C.]  d )
1411brrpss 6296 . . . . . . . . . 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 2580 . . . . . . 7  |-  ( A. d  e.  b  -.  d `' [ C.]  c  <->  A. d  e.  b  -.  c  C.  d )
1817rexbii 2581 . . . . . 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 6302 . . . . . 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 2639 . 2  |-  ( A  e.  Fin  ->  A. b  e.  ~P  ~P A ( ( b  =/=  (/)  /\ [ C.]  Or  b )  ->  U. b  e.  b ) )
25 isfin2 7936 . 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 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165    C. wpss 3166   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    Or wor 4329    Fr wfr 4365   `'ccnv 4704   [ C.] crpss 6292   Fincfn 6879  FinIIcfin2 7921
This theorem is referenced by:  fin1a2s  8056  fin1a2  8057  finngch  8293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rpss 6293  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fin2 7928
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