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Theorem fin23lem23 7952
Description: Lemma for fin23lem22 7953. (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 7951 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E. j  e.  S  ( j  i^i  S
)  ~~  i )
2 ensym 6910 . . . . . 6  |-  ( ( a  i^i  S ) 
~~  i  ->  i  ~~  ( a  i^i  S
) )
3 entr 6913 . . . . . 6  |-  ( ( ( j  i^i  S
)  ~~  i  /\  i  ~~  ( a  i^i 
S ) )  -> 
( j  i^i  S
)  ~~  ( a  i^i  S ) )
42, 3sylan2 460 . . . . 5  |-  ( ( ( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  ( j  i^i  S
)  ~~  ( a  i^i  S ) )
5 simpl 443 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  S  C_  om )
6 simprl 732 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  S )
75, 6sseldd 3181 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  om )
8 nnfi 7053 . . . . . . . . 9  |-  ( j  e.  om  ->  j  e.  Fin )
9 inss1 3389 . . . . . . . . 9  |-  ( j  i^i  S )  C_  j
10 ssfi 7083 . . . . . . . . 9  |-  ( ( j  e.  Fin  /\  ( j  i^i  S
)  C_  j )  ->  ( j  i^i  S
)  e.  Fin )
118, 9, 10sylancl 643 . . . . . . . 8  |-  ( j  e.  om  ->  (
j  i^i  S )  e.  Fin )
127, 11syl 15 . . . . . . 7  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( j  i^i  S )  e.  Fin )
13 simprr 733 . . . . . . . . 9  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  S )
145, 13sseldd 3181 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  om )
15 nnfi 7053 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
16 inss1 3389 . . . . . . . . 9  |-  ( a  i^i  S )  C_  a
17 ssfi 7083 . . . . . . . . 9  |-  ( ( a  e.  Fin  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  Fin )
1815, 16, 17sylancl 643 . . . . . . . 8  |-  ( a  e.  om  ->  (
a  i^i  S )  e.  Fin )
1914, 18syl 15 . . . . . . 7  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( a  i^i  S )  e.  Fin )
20 nnord 4664 . . . . . . . . . 10  |-  ( j  e.  om  ->  Ord  j )
21 nnord 4664 . . . . . . . . . 10  |-  ( a  e.  om  ->  Ord  a )
22 ordtri2or2 4489 . . . . . . . . . 10  |-  ( ( Ord  j  /\  Ord  a )  ->  (
j  C_  a  \/  a  C_  j ) )
2320, 21, 22syl2an 463 . . . . . . . . 9  |-  ( ( j  e.  om  /\  a  e.  om )  ->  ( j  C_  a  \/  a  C_  j ) )
247, 14, 23syl2anc 642 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( j  C_  a  \/  a  C_  j ) )
25 ssrin 3394 . . . . . . . . 9  |-  ( j 
C_  a  ->  (
j  i^i  S )  C_  ( a  i^i  S
) )
26 ssrin 3394 . . . . . . . . 9  |-  ( a 
C_  j  ->  (
a  i^i  S )  C_  ( j  i^i  S
) )
2725, 26orim12i 502 . . . . . . . 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 15 . . . . . . 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 7950 . . . . . . 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 1182 . . . . . 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 4665 . . . . . . 7  |-  Ord  om
32 fin23lem24 7948 . . . . . . 7  |-  ( ( ( Ord  om  /\  S  C_  om )  /\  ( j  e.  S  /\  a  e.  S
) )  ->  (
( j  i^i  S
)  =  ( a  i^i  S )  <->  j  =  a ) )
3331, 32mpanl1 661 . . . . . 6  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  =  ( a  i^i 
S )  <->  j  =  a ) )
3430, 33bitrd 244 . . . . 5  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
j  i^i  S )  ~~  ( a  i^i  S
)  <->  j  =  a ) )
354, 34syl5ib 210 . . . 4  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  ( (
( j  i^i  S
)  ~~  i  /\  ( a  i^i  S
)  ~~  i )  ->  j  =  a ) )
3635ralrimivva 2635 . . 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 706 . 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 3363 . . . 4  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3938breq1d 4033 . . 3  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
4039reu4 2959 . 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 645 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545    i^i cin 3151    C_ wss 3152   class class class wbr 4023   Ord word 4391   omcom 4656    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  fin23lem22  7953  fin23lem27  7954
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-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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