gtnelioc - Mathbox for Glauco Siliprandi

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Theorem gtnelioc 45489

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

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

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

**Proof of Theorem gtnelioc**
Step	Hyp	Ref	Expression
1		gtnelioc.bltc	. . . 4 ⊢ (𝜑 → 𝐵 < 𝐶)
2		gtnelioc.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	2	rexrd 11224	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelioc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11241	. . . . 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	intn3an3d 1483	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
9		gtnelioc.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
10		elioc2 13370	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	9, 2, 10	syl2anc 584	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 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 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,]cioc 13307

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 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-pre-lttri 11142 ax-pre-lttrn 11143

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 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-ioc 13311

This theorem is referenced by: fourierswlem 46228 fouriersw 46229 etransclem18 46250 etransclem46 46278

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