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

Theorem lenlti 11333
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 11291 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3mp2an 689 1 (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2098   class class class wbr 5139  cr 11106   < clt 11247  cle 11248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-xr 11251  df-le 11253
This theorem is referenced by:  ltnlei  11334  hashgt12el  14383  hashgt12el2  14384  georeclim  15820  geoisumr  15826  divalglem6  16344  umgrislfupgrlem  28876  ballotlem4  34017  signswch  34092  limsup10ex  45035
  Copyright terms: Public domain W3C validator