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Mathbox for Glauco Siliprandi |
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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 40530 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1496 | 1 ⊢ (𝜑 → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 ℝ*cxr 10390 < clt 10391 [,)cico 12465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-xr 10395 df-ico 12469 |
This theorem is referenced by: icomnfinre 40574 xlimmnfvlem1 40853 |
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