![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > icoltubd | Structured version Visualization version GIF version |
Description: An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
icoltubd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
icoltubd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
icoltubd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Ref | Expression |
---|---|
icoltubd | ⊢ (𝜑 → 𝐶 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoltubd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | icoltubd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | icoltubd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) | |
4 | icoltub 44156 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5147 (class class class)co 7404 ℝ*cxr 11243 < clt 11244 [,)cico 13322 |
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 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 |
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 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-xr 11248 df-ico 13326 |
This theorem is referenced by: icomnfinre 44200 xlimmnfvlem1 44483 |
Copyright terms: Public domain | W3C validator |