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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 10695 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 class class class wbr 5057 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-le 10669 |
This theorem is referenced by: supxrub 12705 infxrlb 12715 ixxub 12747 ixxlb 12748 supicclub2 12877 radcnvle 24935 xrge0infssd 30411 infxrge0lb 30414 liminflbuz2 41972 icccncfext 42046 |
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