Theorem nltled 11284

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 11280	. 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 5095  ℝcr 11027   < clt 11168   ≤ cle 11169

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 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 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 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-xr 11172  df-le 11174

This theorem is referenced by:  dedekind  11297  suprub  12104  infrelb  12128  suprzub  12858  prodge0rd  13020  seqf1olem1  13966  bitsfzolem  16363  bitsmod  16365  reconnlem2  24732  ioombl1lem4  25478  dgrub  26155  dgrlb  26157  suppssnn0  32763  constrsqrtcl  33745  1smat1  33770  sn-suprubd  42467  imo72b2  44145  dvbdfbdioolem2  45911  stoweidlem14  45996  fourierdlem10  46099  fourierdlem12  46101  fourierdlem20  46109  fourierdlem24  46113  fourierdlem50  46138  fourierdlem54  46142  fourierdlem63  46151  fourierdlem65  46153  fourierdlem75  46163  fourierdlem79  46167  fouriersw  46213  etransclem3  46219  etransclem7  46223  etransclem10  46226  etransclem15  46231  etransclem20  46236  etransclem21  46237  etransclem22  46238  etransclem24  46240  etransclem25  46241  etransclem27  46243  etransclem32  46248