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Theorem ltord1 10167
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 10166 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
8 eqeq1 2465 . . . . . . . 8  |-  ( x  =  C  ->  (
x  =  D  <->  C  =  D ) )
92eqeq1d 2463 . . . . . . . 8  |-  ( x  =  C  ->  ( A  =  N  <->  M  =  N ) )
108, 9imbi12d 326 . . . . . . 7  |-  ( x  =  C  ->  (
( x  =  D  ->  A  =  N )  <->  ( C  =  D  ->  M  =  N ) ) )
1110, 3vtoclg 3118 . . . . . 6  |-  ( C  e.  S  ->  ( C  =  D  ->  M  =  N ) )
1211ad2antrl 739 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  ->  M  =  N ) )
131, 3, 2, 4, 5, 6ltordlem 10166 . . . . . 6  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1413ancom2s 816 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1512, 14orim12d 854 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  =  D  \/  D  < 
C )  ->  ( M  =  N  \/  N  <  M ) ) )
1615con3d 140 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  ( M  =  N  \/  N  <  M )  ->  -.  ( C  =  D  \/  D  <  C ) ) )
175ralrimiva 2813 . . . . . 6  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
182eleq1d 2523 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1918rspccva 3160 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2017, 19sylan 478 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
213eleq1d 2523 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3160 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2317, 22sylan 478 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2420, 23anim12dan 853 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  e.  RR  /\  N  e.  RR ) )
25 axlttri 9730 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
2624, 25syl 17 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
274sseli 3439 . . . . 5  |-  ( C  e.  S  ->  C  e.  RR )
284sseli 3439 . . . . 5  |-  ( D  e.  S  ->  D  e.  RR )
29 axlttri 9730 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3027, 28, 29syl2an 484 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3130adantl 472 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3216, 26, 313imtr4d 276 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  ->  C  <  D ) )
337, 32impbid 195 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  M  <  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748    C_ wss 3415   class class class wbr 4415   RRcr 9563    < clt 9700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-pre-lttri 9638
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-ltxr 9705
This theorem is referenced by:  leord1  10168  ltord2  10170  ltexp2  12357  eflt  14219  tanord1  23534  tanord  23535  monotuz  35833  monotoddzzfi  35834  rpexpmord  35840
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