Theorem nltled 11272

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 11268	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113   class class class wbr 5095  ℝcr 11014   < clt 11155   ≤ cle 11156

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-xr 11159  df-le 11161

This theorem is referenced by:  dedekind  11285  suprub  12092  infrelb  12116  suprzub  12841  prodge0rd  13003  seqf1olem1  13952  bitsfzolem  16349  bitsmod  16351  reconnlem2  24746  ioombl1lem4  25492  dgrub  26169  dgrlb  26171  suppssnn0  32794  constrsqrtcl  33815  1smat1  33840  sn-suprubd  42615  imo72b2  44292  dvbdfbdioolem2  46054  stoweidlem14  46139  fourierdlem10  46242  fourierdlem12  46244  fourierdlem20  46252  fourierdlem24  46256  fourierdlem50  46281  fourierdlem54  46285  fourierdlem63  46294  fourierdlem65  46296  fourierdlem75  46306  fourierdlem79  46310  fouriersw  46356  etransclem3  46362  etransclem7  46366  etransclem10  46369  etransclem15  46374  etransclem20  46379  etransclem21  46380  etransclem22  46381  etransclem24  46383  etransclem25  46384  etransclem27  46386  etransclem32  46391