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

Theorem nltled 11364
Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ltd.1 (𝜑𝐴 ∈ ℝ)
ltd.2 (𝜑𝐵 ∈ ℝ)
nltled.1 (𝜑 → ¬ 𝐵 < 𝐴)
Assertion
Ref Expression
nltled (𝜑𝐴𝐵)

Proof of Theorem nltled
StepHypRef Expression
1 nltled.1 . 2 (𝜑 → ¬ 𝐵 < 𝐴)
2 ltd.1 . . 3 (𝜑𝐴 ∈ ℝ)
3 ltd.2 . . 3 (𝜑𝐵 ∈ ℝ)
42, 3lenltd 11360 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
51, 4mpbird 257 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107   class class class wbr 5149  cr 11109   < clt 11248  cle 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-xr 11252  df-le 11254
This theorem is referenced by:  dedekind  11377  suprub  12175  infrelb  12199  suprzub  12923  prodge0rd  13081  seqf1olem1  14007  bitsfzolem  16375  bitsmod  16377  reconnlem2  24343  ioombl1lem4  25078  dgrub  25748  dgrlb  25750  suppssnn0  32048  1smat1  32815  metakunt28  41060  metakunt30  41062  imo72b2  42972  dvbdfbdioolem2  44693  stoweidlem14  44778  fourierdlem10  44881  fourierdlem12  44883  fourierdlem20  44891  fourierdlem24  44895  fourierdlem50  44920  fourierdlem54  44924  fourierdlem63  44933  fourierdlem65  44935  fourierdlem75  44945  fourierdlem79  44949  fouriersw  44995  etransclem3  45001  etransclem7  45005  etransclem10  45008  etransclem15  45013  etransclem20  45018  etransclem21  45019  etransclem22  45020  etransclem24  45022  etransclem25  45023  etransclem27  45025  etransclem32  45030
  Copyright terms: Public domain W3C validator