Theorem nltled 11263

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 11259	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   class class class wbr 5091  ℝcr 11005   < clt 11146   ≤ cle 11147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-xr 11150  df-le 11152

This theorem is referenced by:  dedekind  11276  suprub  12083  infrelb  12107  suprzub  12837  prodge0rd  12999  seqf1olem1  13948  bitsfzolem  16345  bitsmod  16347  reconnlem2  24744  ioombl1lem4  25490  dgrub  26167  dgrlb  26169  suppssnn0  32785  constrsqrtcl  33790  1smat1  33815  sn-suprubd  42533  imo72b2  44211  dvbdfbdioolem2  45973  stoweidlem14  46058  fourierdlem10  46161  fourierdlem12  46163  fourierdlem20  46171  fourierdlem24  46175  fourierdlem50  46200  fourierdlem54  46204  fourierdlem63  46213  fourierdlem65  46215  fourierdlem75  46225  fourierdlem79  46229  fouriersw  46275  etransclem3  46281  etransclem7  46285  etransclem10  46288  etransclem15  46293  etransclem20  46298  etransclem21  46299  etransclem22  46300  etransclem24  46302  etransclem25  46303  etransclem27  46305  etransclem32  46310