![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nleltd | Structured version Visualization version GIF version |
Description: 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
nleltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
nleltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
nleltd.3 | ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴) |
Ref | Expression |
---|---|
nleltd | ⊢ (𝜑 → 𝐴 < 𝐵) |
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 11405 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
5 | 1, 4 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 class class class wbr 5147 ℝcr 11151 < clt 11292 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-xr 11296 df-le 11298 |
This theorem is referenced by: limsup10exlem 45727 |
Copyright terms: Public domain | W3C validator |