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Mathbox for Glauco Siliprandi |
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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 12606 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1352 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 class class class wbr 4925 (class class class)co 6974 ℝ*cxr 10471 ≤ cle 10473 [,]cicc 12555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-iota 6149 df-fun 6187 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-xr 10476 df-icc 12559 |
This theorem is referenced by: sqrlearg 41294 |
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