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 12826 | . . 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 12795 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
7 | 5, 4, 1, 6 | syl3anc 1367 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
8 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ*) |
9 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ*) |
10 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ*) |
11 | iccgelb 12796 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
12 | 5, 4, 1, 11 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
14 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶) | |
15 | xrltnle 10710 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) | |
16 | 3, 4, 15 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
18 | 14, 17 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 < 𝐵) |
19 | 8, 9, 10, 13, 18 | elicod 12790 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐴[,)𝐵)) |
20 | eliccnelico.nel | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) | |
21 | 20 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐶 ∈ (𝐴[,)𝐵)) |
22 | 19, 21 | condan 816 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
23 | 3, 4, 7, 22 | xrletrid 12551 | 1 ⊢ (𝜑 → 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 [,)cico 12743 [,]cicc 12744 |
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-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ico 12747 df-icc 12748 |
This theorem is referenced by: sge0f1o 42671 |
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