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Theorem fin23lem23 8706
Description: Lemma for fin23lem22 8707. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
fin23lem23  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
Distinct variable group:    i, j, S

Proof of Theorem fin23lem23
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fin23lem26 8705 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E. j  e.  S  ( j  i^i  S
)  ~~  i )
2 ensym 7564 . . . . . 6  |-  ( ( a  i^i  S ) 
~~  i  ->  i  ~~  ( a  i^i  S
) )
3 entr 7567 . . . . . 6  |-  ( ( ( j  i^i  S
)  ~~  i  /\  i  ~~  ( a  i^i 
S ) )  -> 
( j  i^i  S
)  ~~  ( a  i^i  S ) )
42, 3sylan2 474 . . . . 5  |-  ( ( ( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  ( j  i^i  S
)  ~~  ( a  i^i  S ) )
5 simpl 457 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  S  C_  om )
6 simprl 755 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  S )
75, 6sseldd 3505 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  om )
8 nnfi 7710 . . . . . . . . 9  |-  ( j  e.  om  ->  j  e.  Fin )
9 inss1 3718 . . . . . . . . 9  |-  ( j  i^i  S )  C_  j
10 ssfi 7740 . . . . . . . . 9  |-  ( ( j  e.  Fin  /\  ( j  i^i  S
)  C_  j )  ->  ( j  i^i  S
)  e.  Fin )
118, 9, 10sylancl 662 . . . . . . . 8  |-  ( j  e.  om  ->  (
j  i^i  S )  e.  Fin )
127, 11syl 16 . . . . . . 7  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( j  i^i  S )  e.  Fin )
13 simprr 756 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  S )
145, 13sseldd 3505 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  om )
15 nnfi 7710 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
16 inss1 3718 . . . . . . . . 9  |-  ( a  i^i  S )  C_  a
17 ssfi 7740 . . . . . . . . 9  |-  ( ( a  e.  Fin  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  Fin )
1815, 16, 17sylancl 662 . . . . . . . 8  |-  ( a  e.  om  ->  (
a  i^i  S )  e.  Fin )
1914, 18syl 16 . . . . . . 7  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( a  i^i  S )  e.  Fin )
20 nnord 6692 . . . . . . . . . 10  |-  ( j  e.  om  ->  Ord  j )
21 nnord 6692 . . . . . . . . . 10  |-  ( a  e.  om  ->  Ord  a )
22 ordtri2or2 4974 . . . . . . . . . 10  |-  ( ( Ord  j  /\  Ord  a )  ->  (
j  C_  a  \/  a  C_  j ) )
2320, 21, 22syl2an 477 . . . . . . . . 9  |-  ( ( j  e.  om  /\  a  e.  om )  ->  ( j  C_  a  \/  a  C_  j ) )
247, 14, 23syl2anc 661 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( j  C_  a  \/  a  C_  j ) )
25 ssrin 3723 . . . . . . . . 9  |-  ( j 
C_  a  ->  (
j  i^i  S )  C_  ( a  i^i  S
) )
26 ssrin 3723 . . . . . . . . 9  |-  ( a 
C_  j  ->  (
a  i^i  S )  C_  ( j  i^i  S
) )
2725, 26orim12i 516 . . . . . . . 8  |-  ( ( j  C_  a  \/  a  C_  j )  -> 
( ( j  i^i 
S )  C_  (
a  i^i  S )  \/  ( a  i^i  S
)  C_  ( j  i^i  S ) ) )
2824, 27syl 16 . . . . . . 7  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  C_  ( a  i^i  S
)  \/  ( a  i^i  S )  C_  ( j  i^i  S
) ) )
29 fin23lem25 8704 . . . . . . 7  |-  ( ( ( j  i^i  S
)  e.  Fin  /\  ( a  i^i  S
)  e.  Fin  /\  ( ( j  i^i 
S )  C_  (
a  i^i  S )  \/  ( a  i^i  S
)  C_  ( j  i^i  S ) ) )  ->  ( ( j  i^i  S )  ~~  ( a  i^i  S
)  <->  ( j  i^i 
S )  =  ( a  i^i  S ) ) )
3012, 19, 28, 29syl3anc 1228 . . . . . 6  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  ~~  ( a  i^i  S
)  <->  ( j  i^i 
S )  =  ( a  i^i  S ) ) )
31 ordom 6693 . . . . . . 7  |-  Ord  om
32 fin23lem24 8702 . . . . . . 7  |-  ( ( ( Ord  om  /\  S  C_  om )  /\  ( j  e.  S  /\  a  e.  S
) )  ->  (
( j  i^i  S
)  =  ( a  i^i  S )  <->  j  =  a ) )
3331, 32mpanl1 680 . . . . . 6  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  =  ( a  i^i 
S )  <->  j  =  a ) )
3430, 33bitrd 253 . . . . 5  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  ~~  ( a  i^i  S
)  <->  j  =  a ) )
354, 34syl5ib 219 . . . 4  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  j  =  a ) )
3635ralrimivva 2885 . . 3  |-  ( S 
C_  om  ->  A. j  e.  S  A. a  e.  S  ( (
( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  j  =  a ) )
3736ad2antrr 725 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  A. j  e.  S  A. a  e.  S  ( ( ( j  i^i  S )  ~~  i  /\  ( a  i^i 
S )  ~~  i
)  ->  j  =  a ) )
38 ineq1 3693 . . . 4  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3938breq1d 4457 . . 3  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
4039reu4 3297 . 2  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  i  <->  ( E. j  e.  S  (
j  i^i  S )  ~~  i  /\  A. j  e.  S  A. a  e.  S  ( (
( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  j  =  a ) ) )
411, 37, 40sylanbrc 664 1  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   E!wreu 2816    i^i cin 3475    C_ wss 3476   class class class wbr 4447   Ord word 4877   omcom 6684    ~~ cen 7513   Fincfn 7516
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520
This theorem is referenced by:  fin23lem22  8707  fin23lem27  8708
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