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Theorem ltordlem 8453
Description: Lemma for eqord1 8454. (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
ltordlem  |-  ( (
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 ltordlem
StepHypRef Expression
1 ltord.6 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
21ralrimivva 2569 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x  <  y  ->  A  <  B ) )
3 breq1 4018 . . . 4  |-  ( x  =  C  ->  (
x  <  y  <->  C  <  y ) )
4 ltord.2 . . . . 5  |-  ( x  =  C  ->  A  =  M )
54breq1d 4025 . . . 4  |-  ( x  =  C  ->  ( A  <  B  <->  M  <  B ) )
63, 5imbi12d 234 . . 3  |-  ( x  =  C  ->  (
( x  <  y  ->  A  <  B )  <-> 
( C  <  y  ->  M  <  B ) ) )
7 breq2 4019 . . . 4  |-  ( y  =  D  ->  ( C  <  y  <->  C  <  D ) )
8 eqeq1 2194 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  D  <->  y  =  D ) )
9 ltord.1 . . . . . . . 8  |-  ( x  =  y  ->  A  =  B )
109eqeq1d 2196 . . . . . . 7  |-  ( x  =  y  ->  ( A  =  N  <->  B  =  N ) )
118, 10imbi12d 234 . . . . . 6  |-  ( x  =  y  ->  (
( x  =  D  ->  A  =  N )  <->  ( y  =  D  ->  B  =  N ) ) )
12 ltord.3 . . . . . 6  |-  ( x  =  D  ->  A  =  N )
1311, 12chvarv 1947 . . . . 5  |-  ( y  =  D  ->  B  =  N )
1413breq2d 4027 . . . 4  |-  ( y  =  D  ->  ( M  <  B  <->  M  <  N ) )
157, 14imbi12d 234 . . 3  |-  ( y  =  D  ->  (
( C  <  y  ->  M  <  B )  <-> 
( C  <  D  ->  M  <  N ) ) )
166, 15rspc2v 2866 . 2  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( A. x  e.  S  A. y  e.  S  ( x  < 
y  ->  A  <  B )  ->  ( C  <  D  ->  M  <  N ) ) )
172, 16mpan9 281 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   A.wral 2465    C_ wss 3141   class class class wbr 4015   RRcr 7824    < clt 8006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016
This theorem is referenced by:  eqord1  8454
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