Theorem xrnltled 11043

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 11041	. 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	1, 4	mpbird 256	1 ⊢ (𝜑 → 𝐴 ≤ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   class class class wbr 5074  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-le 11015

This theorem is referenced by:  supxrub  13058  infxrlb  13068  ixxub  13100  ixxlb  13101  supicclub2  13236  radcnvle  25579  xrge0infssd  31084  infxrge0lb  31087  liminflbuz2  43356  icccncfext  43428