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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 10701 | . 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 2110 class class class wbr 5058 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-le 10675 |
This theorem is referenced by: supxrub 12711 infxrlb 12721 ixxub 12753 ixxlb 12754 supicclub2 12883 radcnvle 25002 xrge0infssd 30479 infxrge0lb 30482 liminflbuz2 42089 icccncfext 42163 |
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