Theorem nleltd 40171

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 10368	. 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
5	1, 4	mpbird 247	1 ⊢ (𝜑 → 𝐴 < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2131   class class class wbr 4796  ℝcr 10119   < clt 10258   ≤ cle 10259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-cnv 5266  df-xr 10262  df-le 10264

This theorem is referenced by:  limsup10exlem  40499