Theorem xrltnled 11204

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
xrltnled.1	⊢ (𝜑 → 𝐴 ∈ ℝ*)
xrltnled.2	⊢ (𝜑 → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
xrltnled	⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Proof of Theorem xrltnled

Step	Hyp	Ref	Expression
1		xrltnled.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		xrltnled.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
3		xrltnle 11203	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2114   class class class wbr 5086  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171

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 5231  ax-pr 5370

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-le 11176

This theorem is referenced by:  extdgfialglem1  33852  infxrbnd2  45816  infleinflem2  45818  xrralrecnnge  45837  qinioo  45983  limsuppnflem  46156  limsupre2lem  46170  meaiuninc3v  46930  ovolval4lem1  47095  preimagelt  47145  preimalegt  47146