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Theorem ltord1 9878
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 9877 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
8 eqeq1 2449 . . . . . . . 8  |-  ( x  =  C  ->  (
x  =  D  <->  C  =  D ) )
92eqeq1d 2451 . . . . . . . 8  |-  ( x  =  C  ->  ( A  =  N  <->  M  =  N ) )
108, 9imbi12d 320 . . . . . . 7  |-  ( x  =  C  ->  (
( x  =  D  ->  A  =  N )  <->  ( C  =  D  ->  M  =  N ) ) )
1110, 3vtoclg 3042 . . . . . 6  |-  ( C  e.  S  ->  ( C  =  D  ->  M  =  N ) )
1211ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  ->  M  =  N ) )
131, 3, 2, 4, 5, 6ltordlem 9877 . . . . . 6  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1413ancom2s 800 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1512, 14orim12d 834 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  =  D  \/  D  < 
C )  ->  ( M  =  N  \/  N  <  M ) ) )
1615con3d 133 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  ( M  =  N  \/  N  <  M )  ->  -.  ( C  =  D  \/  D  <  C ) ) )
175ralrimiva 2811 . . . . . 6  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
182eleq1d 2509 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1918rspccva 3084 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2017, 19sylan 471 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
213eleq1d 2509 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3084 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2317, 22sylan 471 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2420, 23anim12dan 833 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  e.  RR  /\  N  e.  RR ) )
25 axlttri 9458 . . . 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 3364 . . . . 5  |-  ( C  e.  S  ->  C  e.  RR )
284sseli 3364 . . . . 5  |-  ( D  e.  S  ->  D  e.  RR )
29 axlttri 9458 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3027, 28, 29syl2an 477 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3130adantl 466 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3216, 26, 313imtr4d 268 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  ->  C  <  D ) )
337, 32impbid 191 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 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   class class class wbr 4304   RRcr 9293    < clt 9430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-pre-lttri 9368
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  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 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-ltxr 9435
This theorem is referenced by:  leord1  9879  ltord2  9881  ltexp2  11929  eflt  13413  tanord1  22005  tanord  22006  monotuz  29294  monotoddzzfi  29295  rpexpmord  29301
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