gtnelicc - Mathbox for Glauco Siliprandi

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Theorem gtnelicc 45930

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 11195	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelicc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11212	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	3, 4, 5	syl2anc 585	. . . 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 13366	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	9, 3, 4, 10	syl3anc 1374	. 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 2114 class class class wbr 5085 (class class class)co 7367 ℝcr 11037 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 [,]cicc 13301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-xr 11183 df-le 11185 df-icc 13305

This theorem is referenced by: fourierdlem103 46637

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