Theorem nltled 11324

Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
ltd.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ltd.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
nltled.1	⊢ (𝜑 → ¬ 𝐵 < 𝐴)

Assertion

Ref	Expression
nltled	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Proof of Theorem nltled

Step	Hyp	Ref	Expression
1		nltled.1	. 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
2		ltd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		ltd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	2, 3	lenltd 11320	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   class class class wbr 5107  ℝcr 11067   < clt 11208   ≤ cle 11209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-xr 11212  df-le 11214

This theorem is referenced by:  dedekind  11337  suprub  12144  infrelb  12168  suprzub  12898  prodge0rd  13060  seqf1olem1  14006  bitsfzolem  16404  bitsmod  16406  reconnlem2  24716  ioombl1lem4  25462  dgrub  26139  dgrlb  26141  suppssnn0  32730  constrsqrtcl  33769  1smat1  33794  sn-suprubd  42482  imo72b2  44161  dvbdfbdioolem2  45927  stoweidlem14  46012  fourierdlem10  46115  fourierdlem12  46117  fourierdlem20  46125  fourierdlem24  46129  fourierdlem50  46154  fourierdlem54  46158  fourierdlem63  46167  fourierdlem65  46169  fourierdlem75  46179  fourierdlem79  46183  fouriersw  46229  etransclem3  46235  etransclem7  46239  etransclem10  46242  etransclem15  46247  etransclem20  46252  etransclem21  46253  etransclem22  46254  etransclem24  46256  etransclem25  46257  etransclem27  46259  etransclem32  46264