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Theorem ltordlem 9314
Description: Lemma for ltord1 9315. (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 2648 . 2  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x  <  y  ->  A  <  B ) )
3 breq1 4042 . . . 4  |-  ( x  =  C  ->  (
x  <  y  <->  C  <  y ) )
4 ltord.2 . . . . 5  |-  ( x  =  C  ->  A  =  M )
54breq1d 4049 . . . 4  |-  ( x  =  C  ->  ( A  <  B  <->  M  <  B ) )
63, 5imbi12d 311 . . 3  |-  ( x  =  C  ->  (
( x  <  y  ->  A  <  B )  <-> 
( C  <  y  ->  M  <  B ) ) )
7 breq2 4043 . . . 4  |-  ( y  =  D  ->  ( C  <  y  <->  C  <  D ) )
8 eqeq1 2302 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  D  <->  y  =  D ) )
9 ltord.1 . . . . . . . 8  |-  ( x  =  y  ->  A  =  B )
109eqeq1d 2304 . . . . . . 7  |-  ( x  =  y  ->  ( A  =  N  <->  B  =  N ) )
118, 10imbi12d 311 . . . . . 6  |-  ( x  =  y  ->  (
( x  =  D  ->  A  =  N )  <->  ( y  =  D  ->  B  =  N ) ) )
12 ltord.3 . . . . . 6  |-  ( x  =  D  ->  A  =  N )
1311, 12chvarv 1966 . . . . 5  |-  ( y  =  D  ->  B  =  N )
1413breq2d 4051 . . . 4  |-  ( y  =  D  ->  ( M  <  B  <->  M  <  N ) )
157, 14imbi12d 311 . . 3  |-  ( y  =  D  ->  (
( C  <  y  ->  M  <  B )  <-> 
( C  <  D  ->  M  <  N ) ) )
166, 15rspc2v 2903 . 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 455 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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:  ltord1  9315
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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