Theorem nleltd 45367

Description: 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
nleltd.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
nleltd.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
nleltd.3	⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴)

Assertion

Ref	Expression
nleltd	⊢ (𝜑 → 𝐴 < 𝐵)

Proof of Theorem nleltd

Step	Hyp	Ref	Expression
1		nleltd.3	. 2 ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴)
2		nleltd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		nleltd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	2, 3	ltnled 11437	. 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108   class class class wbr 5166  ℝcr 11183   < 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-xr 11328  df-le 11330

This theorem is referenced by:  limsup10exlem  45693