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Theorem lenlti 9721
Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
Hypotheses
Ref Expression
lt.1  |-  A  e.  RR
lt.2  |-  B  e.  RR
Assertion
Ref Expression
lenlti  |-  ( A  <_  B  <->  -.  B  <  A )

Proof of Theorem lenlti
StepHypRef Expression
1 lt.1 . 2  |-  A  e.  RR
2 lt.2 . 2  |-  B  e.  RR
3 lenlt 9680 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
41, 2, 3mp2an 672 1  |-  ( A  <_  B  <->  -.  B  <  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1819   class class class wbr 4456   RRcr 9508    < clt 9645    <_ cle 9646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-xr 9649  df-le 9651
This theorem is referenced by:  ltnlei  9722  ltadd2iOLD  9733  hashgt12el  12484  hashgt12el2  12485  georeclim  13692  geoisumr  13698  divalglem6  14067  konigsberg  25113  ballotlem4  28612  signswch  28693
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