Theorem xrltnled 45374

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 11328	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2108   class class class wbr 5143  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-le 11301

This theorem is referenced by:  infxrbnd2  45380  infleinflem2  45382  xrralrecnnge  45401  qinioo  45548  limsuppnflem  45725  limsupre2lem  45739  meaiuninc3v  46499  ovolval4lem1  46664  preimagelt  46714  preimalegt  46715