Theorem xrnltled 11181

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 11178	. 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  ℝ^*cxr 11145   < 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-le 11152

This theorem is referenced by:  supxrub  13223  infxrlb  13234  ixxub  13266  ixxlb  13267  supicclub2  13404  radcnvle  26356  xrge0infssd  32744  infxrge0lb  32747  pimxrneun  45534  liminflbuz2  45861  icccncfext  45933