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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccleubd | Structured version Visualization version GIF version |
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
iccleubd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
iccleubd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
iccleubd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Ref | Expression |
---|---|
iccleubd | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccleubd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | iccleubd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | iccleubd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
4 | iccleub 12955 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝ*cxr 10831 ≤ cle 10833 [,]cicc 12903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-xr 10836 df-icc 12907 |
This theorem is referenced by: sqrlearg 42707 |
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