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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > icogelbd | Structured version Visualization version GIF version |
Description: An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
icogelbd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
icogelbd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
icogelbd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Ref | Expression |
---|---|
icogelbd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icogelbd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | icogelbd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | icogelbd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) | |
4 | icogelb 13372 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℝ*cxr 11244 ≤ cle 11246 [,)cico 13323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-xr 11249 df-ico 13327 |
This theorem is referenced by: uzinico 44260 limsupresico 44403 limsupmnflem 44423 liminfresico 44474 liminflelimsuplem 44478 smfliminflem 45533 |
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