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Theorem fin23lem23 8206
Description: Lemma for fin23lem22 8207. (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 8205 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E. j  e.  S  ( j  i^i  S
)  ~~  i )
2 ensym 7156 . . . . . 6  |-  ( ( a  i^i  S ) 
~~  i  ->  i  ~~  ( a  i^i  S
) )
3 entr 7159 . . . . . 6  |-  ( ( ( j  i^i  S
)  ~~  i  /\  i  ~~  ( a  i^i 
S ) )  -> 
( j  i^i  S
)  ~~  ( a  i^i  S ) )
42, 3sylan2 461 . . . . 5  |-  ( ( ( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  ( j  i^i  S
)  ~~  ( a  i^i  S ) )
5 simpl 444 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  S  C_  om )
6 simprl 733 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  S )
75, 6sseldd 3349 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  om )
8 nnfi 7299 . . . . . . . . 9  |-  ( j  e.  om  ->  j  e.  Fin )
9 inss1 3561 . . . . . . . . 9  |-  ( j  i^i  S )  C_  j
10 ssfi 7329 . . . . . . . . 9  |-  ( ( j  e.  Fin  /\  ( j  i^i  S
)  C_  j )  ->  ( j  i^i  S
)  e.  Fin )
118, 9, 10sylancl 644 . . . . . . . 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 734 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  S )
145, 13sseldd 3349 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  om )
15 nnfi 7299 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
16 inss1 3561 . . . . . . . . 9  |-  ( a  i^i  S )  C_  a
17 ssfi 7329 . . . . . . . . 9  |-  ( ( a  e.  Fin  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  Fin )
1815, 16, 17sylancl 644 . . . . . . . 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 4853 . . . . . . . . . 10  |-  ( j  e.  om  ->  Ord  j )
21 nnord 4853 . . . . . . . . . 10  |-  ( a  e.  om  ->  Ord  a )
22 ordtri2or2 4678 . . . . . . . . . 10  |-  ( ( Ord  j  /\  Ord  a )  ->  (
j  C_  a  \/  a  C_  j ) )
2320, 21, 22syl2an 464 . . . . . . . . 9  |-  ( ( j  e.  om  /\  a  e.  om )  ->  ( j  C_  a  \/  a  C_  j ) )
247, 14, 23syl2anc 643 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( j  C_  a  \/  a  C_  j ) )
25 ssrin 3566 . . . . . . . . 9  |-  ( j 
C_  a  ->  (
j  i^i  S )  C_  ( a  i^i  S
) )
26 ssrin 3566 . . . . . . . . 9  |-  ( a 
C_  j  ->  (
a  i^i  S )  C_  ( j  i^i  S
) )
2725, 26orim12i 503 . . . . . . . 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 8204 . . . . . . 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 1184 . . . . . 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 4854 . . . . . . 7  |-  Ord  om
32 fin23lem24 8202 . . . . . . 7  |-  ( ( ( Ord  om  /\  S  C_  om )  /\  ( j  e.  S  /\  a  e.  S
) )  ->  (
( j  i^i  S
)  =  ( a  i^i  S )  <->  j  =  a ) )
3331, 32mpanl1 662 . . . . . 6  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  =  ( a  i^i 
S )  <->  j  =  a ) )
3430, 33bitrd 245 . . . . 5  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  ~~  ( a  i^i  S
)  <->  j  =  a ) )
354, 34syl5ib 211 . . . 4  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  j  =  a ) )
3635ralrimivva 2798 . . 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 707 . 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 3535 . . . 4  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3938breq1d 4222 . . 3  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
4039reu4 3128 . 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 646 1  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707    i^i cin 3319    C_ wss 3320   class class class wbr 4212   Ord word 4580   omcom 4845    ~~ cen 7106   Fincfn 7109
This theorem is referenced by:  fin23lem22  8207  fin23lem27  8208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113
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