Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliccelicod | Structured version Visualization version GIF version |
Description: A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
eliccelicod.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
eliccelicod.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
eliccelicod.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
eliccelicod.d | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
Ref | Expression |
---|---|
eliccelicod | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccelicod.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | eliccelicod.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | eliccelicod.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
4 | eliccxr 13155 | . . 3 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ∈ ℝ*) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
6 | iccgelb 13123 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
7 | 1, 2, 3, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
8 | iccleub 13122 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
9 | 1, 2, 3, 8 | syl3anc 1370 | . . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
10 | eliccelicod.d | . . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
11 | 5, 2, 9, 10 | xrleneltd 42821 | . 2 ⊢ (𝜑 → 𝐶 < 𝐵) |
12 | 1, 2, 5, 7, 11 | elicod 13117 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 (class class class)co 7268 ℝ*cxr 10996 ≤ cle 10998 [,)cico 13069 [,]cicc 13070 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-pre-lttri 10933 ax-pre-lttrn 10934 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-po 5499 df-so 5500 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7821 df-2nd 7822 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-ico 13073 df-icc 13074 |
This theorem is referenced by: carageniuncl 44020 |
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