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Theorem ltord1 9485
Description: Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ltord.1  |-  ( x  =  y  ->  A  =  B )
ltord.2  |-  ( x  =  C  ->  A  =  M )
ltord.3  |-  ( x  =  D  ->  A  =  N )
ltord.4  |-  S  C_  RR
ltord.5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
ltord.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
Assertion
Ref Expression
ltord1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  M  <  N ) )
Distinct variable groups:    x, B    x, y, C    x, D, y    x, M, y    x, N, y    ph, x, y   
x, S, y
Allowed substitution hints:    A( x, y)    B( y)

Proof of Theorem ltord1
StepHypRef Expression
1 ltord.1 . . 3  |-  ( x  =  y  ->  A  =  B )
2 ltord.2 . . 3  |-  ( x  =  C  ->  A  =  M )
3 ltord.3 . . 3  |-  ( x  =  D  ->  A  =  N )
4 ltord.4 . . 3  |-  S  C_  RR
5 ltord.5 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6ltordlem 9484 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
8 eqeq1 2393 . . . . . . . 8  |-  ( x  =  C  ->  (
x  =  D  <->  C  =  D ) )
92eqeq1d 2395 . . . . . . . 8  |-  ( x  =  C  ->  ( A  =  N  <->  M  =  N ) )
108, 9imbi12d 312 . . . . . . 7  |-  ( x  =  C  ->  (
( x  =  D  ->  A  =  N )  <->  ( C  =  D  ->  M  =  N ) ) )
1110, 3vtoclg 2954 . . . . . 6  |-  ( C  e.  S  ->  ( C  =  D  ->  M  =  N ) )
1211ad2antrl 709 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  ->  M  =  N ) )
131, 3, 2, 4, 5, 6ltordlem 9484 . . . . . 6  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1413ancom2s 778 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1512, 14orim12d 812 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  =  D  \/  D  < 
C )  ->  ( M  =  N  \/  N  <  M ) ) )
1615con3d 127 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  ( M  =  N  \/  N  <  M )  ->  -.  ( C  =  D  \/  D  <  C ) ) )
175ralrimiva 2732 . . . . . 6  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
182eleq1d 2453 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1918rspccva 2994 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2017, 19sylan 458 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
213eleq1d 2453 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 2994 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2317, 22sylan 458 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2420, 23anim12dan 811 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  e.  RR  /\  N  e.  RR ) )
25 axlttri 9080 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
2624, 25syl 16 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
274sseli 3287 . . . . 5  |-  ( C  e.  S  ->  C  e.  RR )
284sseli 3287 . . . . 5  |-  ( D  e.  S  ->  D  e.  RR )
29 axlttri 9080 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3027, 28, 29syl2an 464 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3130adantl 453 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3216, 26, 313imtr4d 260 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  ->  C  <  D ) )
337, 32impbid 184 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  M  <  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   class class class wbr 4153   RRcr 8922    < clt 9053
This theorem is referenced by:  leord1  9486  ltord2  9488  ltexp2  11360  eflt  12645  tanord1  20306  tanord  20307  monotuz  26695  monotoddzzfi  26696  rpexpmord  26702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-pre-lttri 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058
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