ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltletr Unicode version

Theorem ltletr 7821
Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
Assertion
Ref Expression
ltletr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <  C
) )

Proof of Theorem ltletr
StepHypRef Expression
1 simprr 506 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  <_  C )
2 simpl2 970 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  B  e.  RR )
3 simpl3 971 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  C  e.  RR )
4 lenlt 7808 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
52, 3, 4syl2anc 408 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( B  <_  C  <->  -.  C  <  B ) )
61, 5mpbid 146 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  -.  C  <  B )
7 simprl 505 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  B )
8 axltwlin 7800 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
98adantr 274 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( A  <  B  ->  ( A  <  C  \/  C  < 
B ) ) )
107, 9mpd 13 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( A  <  C  \/  C  < 
B ) )
116, 10ecased 1312 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  B  /\  B  <_  C ) )  ->  A  <  C )
1211ex 114 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    \/ wo 682    /\ w3a 947    e. wcel 1465   class class class wbr 3899   RRcr 7587    < clt 7768    <_ cle 7769
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltwlin 7701
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774
This theorem is referenced by:  ltletri  7838  ltletrd  8153  ltleadd  8176  nngt0  8713  nnrecgt0  8726  elnnnn0c  8990  elnnz1  9045  zltp1le  9076  uz3m2nn  9336  ledivge1le  9481  addlelt  9523  zltaddlt1le  9757  elfz1b  9838  elfzodifsumelfzo  9946  ssfzo12bi  9970  cos01gt0  11396  oddge22np1  11505  nn0seqcvgd  11649  coprm  11749
  Copyright terms: Public domain W3C validator