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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliccnelico | Structured version Visualization version GIF version |
Description: An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
eliccnelico.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
eliccnelico.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
eliccnelico.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
eliccnelico.nel | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) |
Ref | Expression |
---|---|
eliccnelico | ⊢ (𝜑 → 𝐶 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccnelico.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
2 | eliccxr 12572 | . . 3 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ∈ ℝ*) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
4 | eliccnelico.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | eliccnelico.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
6 | iccleub 12541 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
7 | 5, 4, 1, 6 | syl3anc 1439 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
8 | 5 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ*) |
9 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ*) |
10 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ*) |
11 | iccgelb 12542 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
12 | 5, 4, 1, 11 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
13 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
14 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶) | |
15 | xrltnle 10444 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) | |
16 | 3, 4, 15 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
17 | 16 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
18 | 14, 17 | mpbird 249 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 < 𝐵) |
19 | 8, 9, 10, 13, 18 | elicod 12536 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐴[,)𝐵)) |
20 | eliccnelico.nel | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) | |
21 | 20 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐶 ∈ (𝐴[,)𝐵)) |
22 | 19, 21 | condan 808 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
23 | 3, 4, 7, 22 | xrletrid 12298 | 1 ⊢ (𝜑 → 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 (class class class)co 6922 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 [,)cico 12489 [,]cicc 12490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-ico 12493 df-icc 12494 |
This theorem is referenced by: sge0f1o 41516 |
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