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 10787 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
5 | 1, 4 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 < clt 10675 ≤ cle 10676 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-xr 10679 df-le 10681 |
This theorem is referenced by: limsup10exlem 42073 |
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