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Theorem lenlti 10036
 Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
Hypotheses
Ref Expression
lt.1 𝐴 ∈ ℝ
lt.2 𝐵 ∈ ℝ
Assertion
Ref Expression
lenlti (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)

Proof of Theorem lenlti
StepHypRef Expression
1 lt.1 . 2 𝐴 ∈ ℝ
2 lt.2 . 2 𝐵 ∈ ℝ
3 lenlt 9995 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3mp2an 704 1 (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∈ wcel 1977   class class class wbr 4583  ℝcr 9814   < clt 9953   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-xr 9957  df-le 9959 This theorem is referenced by:  ltnlei  10037  hashgt12el  13070  hashgt12el2  13071  georeclim  14442  geoisumr  14448  divalglem6  14959  umgrislfupgrlem  25788  konigsberg  26514  ballotlem4  29887  signswch  29964
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