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Theorem lenlti 10195
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 10154 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3mp2an 708 1 (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2030   class class class wbr 4685  cr 9973   < clt 10112  cle 10113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-xr 10116  df-le 10118
This theorem is referenced by:  ltnlei  10196  hashgt12el  13248  hashgt12el2  13249  georeclim  14647  geoisumr  14653  divalglem6  15168  umgrislfupgrlem  26062  ballotlem4  30688  signswch  30766
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