![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 12813 | . . 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 12780 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
7 | 5, 4, 1, 6 | syl3anc 1368 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
8 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ*) |
9 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ*) |
10 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ*) |
11 | iccgelb 12781 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
12 | 5, 4, 1, 11 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
14 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶) | |
15 | xrltnle 10697 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) | |
16 | 3, 4, 15 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
18 | 14, 17 | mpbird 260 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 < 𝐵) |
19 | 8, 9, 10, 13, 18 | elicod 12775 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐴[,)𝐵)) |
20 | eliccnelico.nel | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) | |
21 | 20 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐶 ∈ (𝐴[,)𝐵)) |
22 | 19, 21 | condan 817 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
23 | 3, 4, 7, 22 | xrletrid 12536 | 1 ⊢ (𝜑 → 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 [,)cico 12728 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 df-icc 12733 |
This theorem is referenced by: sge0f1o 43021 |
Copyright terms: Public domain | W3C validator |