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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 13353 | . . 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 13320 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵) | |
7 | 5, 4, 1, 6 | syl3anc 1372 | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
8 | 5 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ*) |
9 | 4 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ*) |
10 | 3 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ*) |
11 | iccgelb 13321 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) | |
12 | 5, 4, 1, 11 | syl3anc 1372 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
13 | 12 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
14 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶) | |
15 | xrltnle 11223 | . . . . . . 7 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) | |
16 | 3, 4, 15 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
17 | 16 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶)) |
18 | 14, 17 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 < 𝐵) |
19 | 8, 9, 10, 13, 18 | elicod 13315 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐴[,)𝐵)) |
20 | eliccnelico.nel | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) | |
21 | 20 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐶 ∈ (𝐴[,)𝐵)) |
22 | 19, 21 | condan 817 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
23 | 3, 4, 7, 22 | xrletrid 13075 | 1 ⊢ (𝜑 → 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 [,)cico 13267 [,]cicc 13268 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ico 13271 df-icc 13272 |
This theorem is referenced by: sge0f1o 44630 |
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