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Theorem fin23lem23 7968
Description: Lemma for fin23lem22 7969. (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 7967 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E. j  e.  S  ( j  i^i  S
)  ~~  i )
2 ensym 6926 . . . . . 6  |-  ( ( a  i^i  S ) 
~~  i  ->  i  ~~  ( a  i^i  S
) )
3 entr 6929 . . . . . 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 3194 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  j  e.  om )
8 nnfi 7069 . . . . . . . . 9  |-  ( j  e.  om  ->  j  e.  Fin )
9 inss1 3402 . . . . . . . . 9  |-  ( j  i^i  S )  C_  j
10 ssfi 7099 . . . . . . . . 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 3194 . . . . . . . 8  |-  ( ( S  C_  om  /\  (
j  e.  S  /\  a  e.  S )
)  ->  a  e.  om )
15 nnfi 7069 . . . . . . . . 9  |-  ( a  e.  om  ->  a  e.  Fin )
16 inss1 3402 . . . . . . . . 9  |-  ( a  i^i  S )  C_  a
17 ssfi 7099 . . . . . . . . 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 4680 . . . . . . . . . 10  |-  ( j  e.  om  ->  Ord  j )
21 nnord 4680 . . . . . . . . . 10  |-  ( a  e.  om  ->  Ord  a )
22 ordtri2or2 4505 . . . . . . . . . 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 3407 . . . . . . . . 9  |-  ( j 
C_  a  ->  (
j  i^i  S )  C_  ( a  i^i  S
) )
26 ssrin 3407 . . . . . . . . 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 7966 . . . . . . 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 4681 . . . . . . 7  |-  Ord  om
32 fin23lem24 7964 . . . . . . 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 2648 . . 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 3376 . . . 4  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3938breq1d 4049 . . 3  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
4039reu4 2972 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    i^i cin 3164    C_ wss 3165   class class class wbr 4039   Ord word 4407   omcom 4672    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  fin23lem22  7969  fin23lem27  7970
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-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  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-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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