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Theorem lenlti 8938
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 8901 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
41, 2, 3mp2an 653 1  |-  ( A  <_  B  <->  -.  B  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    e. wcel 1684   class class class wbr 4023   RRcr 8736    < clt 8867    <_ cle 8868
This theorem is referenced by:  ltnlei  8939  ltadd2i  8950  georeclim  12328  geoisumr  12334  divalglem6  12597  ballotlem4  23057  konigsberg  23911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-xr 8871  df-le 8873
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