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Theorem nleltd 40197
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 10396 . 2 (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))
51, 4mpbird 247 1 (𝜑𝐴 < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2139   class class class wbr 4804  cr 10147   < clt 10286  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-xr 10290  df-le 10292
This theorem is referenced by:  limsup10exlem  40525
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