Theorem xrltnled 44073

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 11281	. 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 205   ∈ wcel 2107   class class class wbr 5149  ℝ^*cxr 11247   < clt 11248   ≤ cle 11249

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-le 11254

This theorem is referenced by:  infxrbnd2  44079  infleinflem2  44081  xrralrecnnge  44100  qinioo  44248  limsuppnflem  44426  limsupre2lem  44440  meaiuninc3v  45200  ovolval4lem1  45365  preimagelt  45415  preimalegt  45416