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 5089  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 5232  ax-nul 5242  ax-pr 5368

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 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  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  24743  ioombl1lem4  25489  dgrub  26166  dgrlb  26168  suppssnn0  32787  constrsqrtcl  33792  1smat1  33817  sn-suprubd  42586  imo72b2  44264  dvbdfbdioolem2  46026  stoweidlem14  46111  fourierdlem10  46214  fourierdlem12  46216  fourierdlem20  46224  fourierdlem24  46228  fourierdlem50  46253  fourierdlem54  46257  fourierdlem63  46266  fourierdlem65  46268  fourierdlem75  46278  fourierdlem79  46282  fouriersw  46328  etransclem3  46334  etransclem7  46338  etransclem10  46341  etransclem15  46346  etransclem20  46351  etransclem21  46352  etransclem22  46353  etransclem24  46355  etransclem25  46356  etransclem27  46358  etransclem32  46363