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Theorem ltleletr 7870
Description: Transitive law, weaker form of  ( A  < 
B  /\  B  <_  C )  ->  A  <  C. (Contributed by AV, 14-Oct-2018.)
Assertion
Ref Expression
ltleletr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <_  C
) )

Proof of Theorem ltleletr
StepHypRef Expression
1 lttr 7862 . . . . . 6  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
213coml 1189 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
32expcomd 1418 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  <  A  ->  C  <  B ) ) )
4 con3 632 . . . 4  |-  ( ( C  <  A  ->  C  <  B )  -> 
( -.  C  < 
B  ->  -.  C  <  A ) )
53, 4syl6 33 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( -.  C  <  B  ->  -.  C  <  A ) ) )
6 lenlt 7864 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
763adant1 1000 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
8 lenlt 7864 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
983adant2 1001 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
107, 9imbi12d 233 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  <_  C  ->  A  <_  C )  <->  ( -.  C  <  B  ->  -.  C  <  A
) ) )
115, 10sylibrd 168 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( B  <_  C  ->  A  <_  C ) ) )
1211impd 252 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <_  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    e. wcel 1481   class class class wbr 3937   RRcr 7643    < clt 7824    <_ cle 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-lttrn 7758
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830
This theorem is referenced by:  nn0ge2m1nn  9061  lbzbi  9435  iseqf1olemqk  10298
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