Theorem nleltd 45455

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 11328	. 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   class class class wbr 5110  ℝcr 11074   < clt 11215   ≤ cle 11216

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-xr 11219  df-le 11221

This theorem is referenced by:  limsup10exlem  45777