gtnelicc - Mathbox for Glauco Siliprandi

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Theorem gtnelicc 45509

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 11290	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelicc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11307	. . . . 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 13435	. . 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 2109 class class class wbr 5124 (class class class)co 7410 ℝcr 11133 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 [,]cicc 13370

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 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-xr 11278 df-le 11280 df-icc 13374

This theorem is referenced by: fourierdlem103 46218

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