Theorem nleltd 45564

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 11270	. 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
5	1, 4	mpbird 257	1 ⊢ (𝜑 → 𝐴 < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113   class class class wbr 5095  ℝcr 11015   < clt 11156   ≤ cle 11157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-xr 11160  df-le 11162

This theorem is referenced by:  limsup10exlem  45884