Theorem xrnltled 11213

Description: "Not less than" implies "less than or equal to". (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
xrnltled.1	⊢ (𝜑 → 𝐴 ∈ ℝ*)
xrnltled.2	⊢ (𝜑 → 𝐵 ∈ ℝ*)
xrnltled.3	⊢ (𝜑 → ¬ 𝐵 < 𝐴)

Assertion

Ref	Expression
xrnltled	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Proof of Theorem xrnltled

Step	Hyp	Ref	Expression
1		xrnltled.3	. 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
2		xrnltled.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
3		xrnltled.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
4	2, 3	xrlenltd 11210	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   class class class wbr 5100  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-le 11184

This theorem is referenced by:  supxrub  13251  infxrlb  13262  ixxub  13294  ixxlb  13295  supicclub2  13432  radcnvle  26397  xrge0infssd  32852  infxrge0lb  32855  pimxrneun  45846  liminflbuz2  46173  icccncfext  46245