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Mirrors > Home > MPE Home > Th. List > Mathboxes > iocgtlbd | Structured version Visualization version GIF version |
Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
iocgtlbd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
iocgtlbd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
iocgtlbd.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) |
Ref | Expression |
---|---|
iocgtlbd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocgtlbd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | iocgtlbd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | iocgtlbd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) | |
4 | iocgtlb 41784 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝ*cxr 10676 < clt 10677 (,]cioc 12742 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
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 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-xr 10681 df-ioc 12746 |
This theorem is referenced by: xlimpnfvlem1 42124 |
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