Theorem nltled 11409

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 11405	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   class class class wbr 5148  ℝcr 11152   < clt 11293   ≤ cle 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-xr 11297  df-le 11299

This theorem is referenced by:  dedekind  11422  suprub  12227  infrelb  12251  suprzub  12979  prodge0rd  13140  seqf1olem1  14079  bitsfzolem  16468  bitsmod  16470  reconnlem2  24863  ioombl1lem4  25610  dgrub  26288  dgrlb  26290  suppssnn0  32815  1smat1  33765  metakunt28  42214  metakunt30  42216  sn-suprubd  42481  imo72b2  44162  dvbdfbdioolem2  45885  stoweidlem14  45970  fourierdlem10  46073  fourierdlem12  46075  fourierdlem20  46083  fourierdlem24  46087  fourierdlem50  46112  fourierdlem54  46116  fourierdlem63  46125  fourierdlem65  46127  fourierdlem75  46137  fourierdlem79  46141  fouriersw  46187  etransclem3  46193  etransclem7  46197  etransclem10  46200  etransclem15  46205  etransclem20  46210  etransclem21  46211  etransclem22  46212  etransclem24  46214  etransclem25  46215  etransclem27  46217  etransclem32  46222