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Theorem smoord 7028
Description: A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
Assertion
Ref Expression
smoord  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )

Proof of Theorem smoord
StepHypRef Expression
1 smodm2 7018 . . . 4  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
21adantr 463 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  A )
3 simprl 754 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  C  e.  A )
4 ordelord 4889 . . 3  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
52, 3, 4syl2anc 659 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  C )
6 simprr 755 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  D  e.  A )
7 ordelord 4889 . . 3  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
82, 6, 7syl2anc 659 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  D )
9 ordtri3or 4899 . . 3  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  e.  D  \/  C  =  D  \/  D  e.  C ) )
10 simp3 996 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  C  e.  D )
11 smoel2 7026 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( D  e.  A  /\  C  e.  D
) )  ->  ( F `  C )  e.  ( F `  D
) )
1211expr 613 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  D  e.  A )  ->  ( C  e.  D  ->  ( F `  C
)  e.  ( F `
 D ) ) )
1312adantrl 713 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  ->  ( F `  C )  e.  ( F `  D ) ) )
14133impia 1191 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  ( F `  C )  e.  ( F `  D
) )
1510, 142thd 240 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  e.  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
16153expia 1196 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
17 ordirr 4885 . . . . . . . . 9  |-  ( Ord 
C  ->  -.  C  e.  C )
185, 17syl 16 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  -.  C  e.  C )
19183adant3 1014 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  C  e.  C )
20 simp3 996 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  C  =  D )
2119, 20neleqtrd 2566 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  C  e.  D )
22 smofvon2 7019 . . . . . . . . . 10  |-  ( Smo 
F  ->  ( F `  C )  e.  On )
2322ad2antlr 724 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( F `  C )  e.  On )
24 eloni 4877 . . . . . . . . 9  |-  ( ( F `  C )  e.  On  ->  Ord  ( F `  C ) )
25 ordirr 4885 . . . . . . . . 9  |-  ( Ord  ( F `  C
)  ->  -.  ( F `  C )  e.  ( F `  C
) )
2623, 24, 253syl 20 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  -.  ( F `  C )  e.  ( F `  C ) )
27263adant3 1014 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  ( F `  C )  e.  ( F `  C ) )
2820fveq2d 5852 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  ( F `  C )  =  ( F `  D ) )
2927, 28neleqtrd 2566 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  -.  ( F `  C )  e.  ( F `  D ) )
3021, 292falsed 349 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  C  =  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
31303expia 1196 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  =  D  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
3283adant3 1014 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  Ord  D )
33 ordn2lp 4887 . . . . . . . 8  |-  ( Ord 
D  ->  -.  ( D  e.  C  /\  C  e.  D )
)
3432, 33syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( D  e.  C  /\  C  e.  D
) )
35 pm3.2 445 . . . . . . . 8  |-  ( D  e.  C  ->  ( C  e.  D  ->  ( D  e.  C  /\  C  e.  D )
) )
36353ad2ant3 1017 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( C  e.  D  ->  ( D  e.  C  /\  C  e.  D )
) )
3734, 36mtod 177 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  C  e.  D )
3823, 24syl 16 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  Ord  ( F `  C ) )
39383adant3 1014 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  Ord  ( F `  C ) )
40 ordn2lp 4887 . . . . . . . 8  |-  ( Ord  ( F `  C
)  ->  -.  (
( F `  C
)  e.  ( F `
 D )  /\  ( F `  D )  e.  ( F `  C ) ) )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) )
42 smoel2 7026 . . . . . . . . . 10  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  C
) )  ->  ( F `  D )  e.  ( F `  C
) )
4342adantrlr 720 . . . . . . . . 9  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( ( C  e.  A  /\  D  e.  A )  /\  D  e.  C ) )  -> 
( F `  D
)  e.  ( F `
 C ) )
44433impb 1190 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( F `  D )  e.  ( F `  C
) )
45 pm3.21 446 . . . . . . . 8  |-  ( ( F `  D )  e.  ( F `  C )  ->  (
( F `  C
)  e.  ( F `
 D )  -> 
( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) ) )
4644, 45syl 16 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  (
( F `  C
)  e.  ( F `
 D )  -> 
( ( F `  C )  e.  ( F `  D )  /\  ( F `  D )  e.  ( F `  C ) ) ) )
4741, 46mtod 177 . . . . . 6  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  -.  ( F `  C )  e.  ( F `  D ) )
4837, 472falsed 349 . . . . 5  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
)  /\  D  e.  C )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
49483expia 1196 . . . 4  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( D  e.  C  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
5016, 31, 493jaod 1290 . . 3  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( C  e.  D  \/  C  =  D  \/  D  e.  C
)  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
519, 50syl5 32 . 2  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( Ord  C  /\  Ord  D )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) ) )
525, 8, 51mp2and 677 1  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A
) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823   Ord word 4866   Oncon0 4867    Fn wfn 5565   ` cfv 5570   Smo wsmo 7008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-smo 7009
This theorem is referenced by:  smoword  7029  smoiso2  7032
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