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Mirrors > Home > MPE Home > Th. List > xrnltled | 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 |
---|---|
xrnltled.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrnltled.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrnltled.3 | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Ref | Expression |
---|---|
xrnltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnltled.3 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
2 | xrnltled.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
3 | xrnltled.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
4 | 2, 3 | xrlenltd 11179 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 class class class wbr 5103 ℝ*cxr 11146 < clt 11147 ≤ cle 11148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-le 11153 |
This theorem is referenced by: supxrub 13197 infxrlb 13207 ixxub 13239 ixxlb 13240 supicclub2 13375 radcnvle 25730 xrge0infssd 31490 infxrge0lb 31493 pimxrneun 43622 liminflbuz2 43950 icccncfext 44022 |
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