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Theorem ltordlem 10074
Description: Lemma for ltord1 10075. (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 2875 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x  <  y  ->  A  <  B ) )
3 breq1 4442 . . . 4  |-  ( x  =  C  ->  (
x  <  y  <->  C  <  y ) )
4 ltord.2 . . . . 5  |-  ( x  =  C  ->  A  =  M )
54breq1d 4449 . . . 4  |-  ( x  =  C  ->  ( A  <  B  <->  M  <  B ) )
63, 5imbi12d 318 . . 3  |-  ( x  =  C  ->  (
( x  <  y  ->  A  <  B )  <-> 
( C  <  y  ->  M  <  B ) ) )
7 breq2 4443 . . . 4  |-  ( y  =  D  ->  ( C  <  y  <->  C  <  D ) )
8 eqeq1 2458 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  D  <->  y  =  D ) )
9 ltord.1 . . . . . . . 8  |-  ( x  =  y  ->  A  =  B )
109eqeq1d 2456 . . . . . . 7  |-  ( x  =  y  ->  ( A  =  N  <->  B  =  N ) )
118, 10imbi12d 318 . . . . . 6  |-  ( x  =  y  ->  (
( x  =  D  ->  A  =  N )  <->  ( y  =  D  ->  B  =  N ) ) )
12 ltord.3 . . . . . 6  |-  ( x  =  D  ->  A  =  N )
1311, 12chvarv 2019 . . . . 5  |-  ( y  =  D  ->  B  =  N )
1413breq2d 4451 . . . 4  |-  ( y  =  D  ->  ( M  <  B  <->  M  <  N ) )
157, 14imbi12d 318 . . 3  |-  ( y  =  D  ->  (
( C  <  y  ->  M  <  B )  <-> 
( C  <  D  ->  M  <  N ) ) )
166, 15rspc2v 3216 . 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 467 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   class class class wbr 4439   RRcr 9480    < clt 9617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440
This theorem is referenced by:  ltord1  10075
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