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Theorem nleltd 43694
Description: 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
nleltd.1 (𝜑𝐴 ∈ ℝ)
nleltd.2 (𝜑𝐵 ∈ ℝ)
nleltd.3 (𝜑 → ¬ 𝐵𝐴)
Assertion
Ref Expression
nleltd (𝜑𝐴 < 𝐵)

Proof of Theorem nleltd
StepHypRef Expression
1 nleltd.3 . 2 (𝜑 → ¬ 𝐵𝐴)
2 nleltd.1 . . 3 (𝜑𝐴 ∈ ℝ)
3 nleltd.2 . . 3 (𝜑𝐵 ∈ ℝ)
42, 3ltnled 11303 . 2 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
51, 4mpbird 257 1 (𝜑𝐴 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107   class class class wbr 5106  cr 11051   < clt 11190  cle 11191
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-xr 11194  df-le 11196
This theorem is referenced by:  limsup10exlem  44020
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