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Theorem eqord1 10141
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
eqord1  |-  ( (
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 eqord1
StepHypRef Expression
1 ltord.1 . . . 4  |-  ( x  =  y  ->  A  =  B )
2 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
3 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
4 ltord.4 . . . 4  |-  S  C_  RR
5 ltord.5 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6leord1 10140 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
81, 3, 2, 4, 5, 6leord1 10140 . . . 4  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
98ancom2s 809 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
107, 9anbi12d 715 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  <_  D  /\  D  <_  C
)  <->  ( M  <_  N  /\  N  <_  M
) ) )
114sseli 3466 . . . 4  |-  ( C  e.  S  ->  C  e.  RR )
124sseli 3466 . . . 4  |-  ( D  e.  S  ->  D  e.  RR )
13 letri3 9718 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1411, 12, 13syl2an 479 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1514adantl 467 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
165ralrimiva 2846 . . . . 5  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
172eleq1d 2498 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1817rspccva 3187 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
1916, 18sylan 473 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2019adantrr 721 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
213eleq1d 2498 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3187 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2316, 22sylan 473 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2423adantrl 720 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
2520, 24letri3d 9776 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2610, 15, 253bitr4d 288 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   class class class wbr 4426   RRcr 9537    < clt 9674    <_ cle 9675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680
This theorem is referenced by:  eqord2  10144  expcan  12322  ovolicc2lem3  22350  rmyeq0  35509  rmyeq  35510
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