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Theorem lenlti 9695
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 9654 . 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 1762   class class class wbr 4442   RRcr 9482    < clt 9619    <_ cle 9620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-xr 9623  df-le 9625
This theorem is referenced by:  ltnlei  9696  ltadd2i  9706  hashgt12el  12435  hashgt12el2  12436  georeclim  13635  geoisumr  13641  divalglem6  13906  konigsberg  24651  ballotlem4  28065  signswch  28146
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