Theorem nltled 11385

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 11381	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108   class class class wbr 5119  ℝcr 11128   < clt 11269   ≤ cle 11270

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-xr 11273  df-le 11275

This theorem is referenced by:  dedekind  11398  suprub  12203  infrelb  12227  suprzub  12955  prodge0rd  13116  seqf1olem1  14059  bitsfzolem  16453  bitsmod  16455  reconnlem2  24767  ioombl1lem4  25514  dgrub  26191  dgrlb  26193  suppssnn0  32784  constrsqrtcl  33813  1smat1  33835  metakunt28  42245  metakunt30  42247  sn-suprubd  42517  imo72b2  44196  dvbdfbdioolem2  45958  stoweidlem14  46043  fourierdlem10  46146  fourierdlem12  46148  fourierdlem20  46156  fourierdlem24  46160  fourierdlem50  46185  fourierdlem54  46189  fourierdlem63  46198  fourierdlem65  46200  fourierdlem75  46210  fourierdlem79  46214  fouriersw  46260  etransclem3  46266  etransclem7  46270  etransclem10  46273  etransclem15  46278  etransclem20  46283  etransclem21  46284  etransclem22  46285  etransclem24  46287  etransclem25  46288  etransclem27  46290  etransclem32  46295