gtnelicc - Mathbox for Glauco Siliprandi

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Theorem gtnelicc 45548

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
gtnelicc.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
gtnelicc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
gtnelicc.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
gtnelicc.bltc	⊢ (𝜑 → 𝐵 < 𝐶)

**Assertion**
Ref	Expression
gtnelicc	⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

**Proof of Theorem gtnelicc**
Step	Hyp	Ref	Expression
1		gtnelicc.bltc	. . . 4 ⊢ (𝜑 → 𝐵 < 𝐶)
2		gtnelicc.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	2	rexrd 11162	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelicc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11179	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	3, 4, 5	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
7	1, 6	mpbid 232	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
8	7	intnand 488	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		gtnelicc.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
10		elicc4 13313	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	9, 3, 4, 10	syl3anc 1373	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2111 class class class wbr 5089 (class class class)co 7346 ℝcr 11005 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 [,]cicc 13248

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-iota 6437 df-fun 6483 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-xr 11150 df-le 11152 df-icc 13252

This theorem is referenced by: fourierdlem103 46255

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