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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 41869 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5047 (class class class)co 7137 ℝ*cxr 10655 < clt 10656 [,)cico 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6295 df-fun 6338 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-xr 10660 df-ico 12726 |
This theorem is referenced by: icomnfinre 41913 xlimmnfvlem1 42198 |
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