Theorem xrltnled 45048

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 11338	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
4	1, 2, 3	syl2anc 582	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∈ wcel 2100   class class class wbr 5156  ℝ^*cxr 11304   < clt 11305   ≤ cle 11306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-br 5157  df-opab 5219  df-xp 5692  df-cnv 5694  df-le 11311

This theorem is referenced by:  infxrbnd2  45054  infleinflem2  45056  xrralrecnnge  45075  qinioo  45223  limsuppnflem  45401  limsupre2lem  45415  meaiuninc3v  46175  ovolval4lem1  46340  preimagelt  46390  preimalegt  46391