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Theorem ltleletr 7717
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 7709 . . . . . 6  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
213coml 1156 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
32expcomd 1385 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  <  A  ->  C  <  B ) ) )
4 con3 611 . . . 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 7711 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
763adant1 967 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
8 lenlt 7711 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
983adant2 968 . . . 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 930    e. wcel 1448   class class class wbr 3875   RRcr 7499    < clt 7672    <_ cle 7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-lttrn 7609
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678
This theorem is referenced by:  nn0ge2m1nn  8889  lbzbi  9258  iseqf1olemqk  10108
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