MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lenlti Structured version   Visualization version   GIF version

Theorem lenlti 10101
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 10060 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3mp2an 707 1 (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 1987   class class class wbr 4613  cr 9879   < clt 10018  cle 10019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-xr 10022  df-le 10024
This theorem is referenced by:  ltnlei  10102  hashgt12el  13150  hashgt12el2  13151  georeclim  14528  geoisumr  14534  divalglem6  15045  umgrislfupgrlem  25912  ballotlem4  30341  signswch  30418
  Copyright terms: Public domain W3C validator