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| Mirrors > Home > MPE Home > Th. List > nltled | Structured version Visualization version GIF version | ||
| Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| nltled.1 | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
| Ref | Expression |
|---|---|
| nltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nltled.1 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
| 2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | 2, 3 | lenltd 11407 | . 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: dedekind 11424 suprub 12229 infrelb 12253 suprzub 12981 prodge0rd 13142 seqf1olem1 14082 bitsfzolem 16471 bitsmod 16473 reconnlem2 24849 ioombl1lem4 25596 dgrub 26273 dgrlb 26275 suppssnn0 32809 1smat1 33803 metakunt28 42233 metakunt30 42235 sn-suprubd 42504 imo72b2 44185 dvbdfbdioolem2 45944 stoweidlem14 46029 fourierdlem10 46132 fourierdlem12 46134 fourierdlem20 46142 fourierdlem24 46146 fourierdlem50 46171 fourierdlem54 46175 fourierdlem63 46184 fourierdlem65 46186 fourierdlem75 46196 fourierdlem79 46200 fouriersw 46246 etransclem3 46252 etransclem7 46256 etransclem10 46259 etransclem15 46264 etransclem20 46269 etransclem21 46270 etransclem22 46271 etransclem24 46273 etransclem25 46274 etransclem27 46276 etransclem32 46281 |
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