Theorem xrltnled 45278

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 11357	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2108   class class class wbr 5166  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-le 11330

This theorem is referenced by:  infxrbnd2  45284  infleinflem2  45286  xrralrecnnge  45305  qinioo  45453  limsuppnflem  45631  limsupre2lem  45645  meaiuninc3v  46405  ovolval4lem1  46570  preimagelt  46620  preimalegt  46621