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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 13363 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5142 (class class class)co 7394 ℝ*cxr 11231 ≤ cle 11233 [,]cicc 13311 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 |
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-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-xr 11236 df-icc 13315 |
This theorem is referenced by: sqrlearg 44103 |
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