eliccnelico - Mathbox for Glauco Siliprandi

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Theorem eliccnelico 38603

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 38584	. . 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 12100	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
7	5, 4, 1, 6	syl3anc 1318	. 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 12101	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
12	5, 4, 1, 11	syl3anc 1318	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
14		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → ¬ 𝐵 ≤ 𝐶)
15		xrltnle 9984	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
16	3, 4, 15	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → (𝐶 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐶))
18	14, 17	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝐶) → 𝐶 < 𝐵)
19	8, 9, 10, 13, 18	elicod 12095	. . 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 831	. 2 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
23	3, 4, 7, 22	xrletrid 11862	1 ⊢ (𝜑 → 𝐶 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 [,)cico 12048 [,]cicc 12049

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052 df-icc 12053

This theorem is referenced by: sge0f1o 39275

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