| Mathbox for Glauco Siliprandi |
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| 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 11408 | . 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 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-xr 11299 df-le 11301 |
| This theorem is referenced by: limsup10exlem 45787 |
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