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Theorem ltord1 9315
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 9314 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
8 eqeq1 2302 . . . . . . . 8  |-  ( x  =  C  ->  (
x  =  D  <->  C  =  D ) )
92eqeq1d 2304 . . . . . . . 8  |-  ( x  =  C  ->  ( A  =  N  <->  M  =  N ) )
108, 9imbi12d 311 . . . . . . 7  |-  ( x  =  C  ->  (
( x  =  D  ->  A  =  N )  <->  ( C  =  D  ->  M  =  N ) ) )
1110, 3vtoclg 2856 . . . . . 6  |-  ( C  e.  S  ->  ( C  =  D  ->  M  =  N ) )
1211ad2antrl 708 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  ->  M  =  N ) )
131, 3, 2, 4, 5, 6ltordlem 9314 . . . . . 6  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1413ancom2s 777 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1512, 14orim12d 811 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  =  D  \/  D  < 
C )  ->  ( M  =  N  \/  N  <  M ) ) )
1615con3d 125 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  ( M  =  N  \/  N  <  M )  ->  -.  ( C  =  D  \/  D  <  C ) ) )
175ralrimiva 2639 . . . . . 6  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
182eleq1d 2362 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1918rspccva 2896 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2017, 19sylan 457 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
213eleq1d 2362 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 2896 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2317, 22sylan 457 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2420, 23anim12dan 810 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  e.  RR  /\  N  e.  RR ) )
25 axlttri 8910 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
2624, 25syl 15 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
274sseli 3189 . . . . 5  |-  ( C  e.  S  ->  C  e.  RR )
284sseli 3189 . . . . 5  |-  ( D  e.  S  ->  D  e.  RR )
29 axlttri 8910 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3027, 28, 29syl2an 463 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3130adantl 452 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3216, 26, 313imtr4d 259 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  ->  C  <  D ) )
337, 32impbid 183 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   RRcr 8752    < clt 8883
This theorem is referenced by:  leord1  9316  ltord2  9318  ltexp2  11171  eflt  12413  tanord1  19915  tanord  19916  monotuz  27129  monotoddzzfi  27130  rpexpmord  27136
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  ax-resscn 8810  ax-pre-lttri 8827
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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