eliccnelico - Mathbox for Glauco Siliprandi

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Theorem eliccnelico 45530

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.)

**Hypotheses**
Ref	Expression
eliccnelico.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
eliccnelico.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
eliccnelico.c	⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
eliccnelico.nel	⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))

**Assertion**
Ref	Expression
eliccnelico	⊢ (𝜑 → 𝐶 = 𝐵)

**Proof of Theorem eliccnelico**
Step	Hyp	Ref	Expression
1		eliccnelico.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
2		eliccxr 13338	. . 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 13304	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
7	5, 4, 1, 6	syl3anc 1373	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
8	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ^*)
9	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ^*)
10	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ^*)
11		iccgelb 13305	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
12	5, 4, 1, 11	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
14		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶)
15		xrltnle 11182	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
16	3, 4, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
17	16	adantr 480	. . . . 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 13298	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐴[,)𝐵))
20		eliccnelico.nel	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵))
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐶 ∈ (𝐴[,)𝐵))
22	19, 21	condan 817	. 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
23	3, 4, 7, 22	xrletrid 13057	1 ⊢ (𝜑 → 𝐶 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 class class class wbr 5092 (class class class)co 7349 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 [,)cico 13250 [,]cicc 13251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-ico 13254 df-icc 13255

This theorem is referenced by: sge0f1o 46383

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