gtnelioc - Mathbox for Glauco Siliprandi

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Theorem gtnelioc 44923

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 11304	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelioc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11321	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	3, 4, 5	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
7	1, 6	mpbid 231	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
8	7	intn3an3d 1477	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
9		gtnelioc.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
10		elioc2 13429	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	9, 2, 10	syl2anc 582	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 324	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℝcr 11147 ℝ^*cxr 11287 < clt 11288 ≤ cle 11289 (,]cioc 13367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-pre-lttri 11222 ax-pre-lttrn 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-ioc 13371

This theorem is referenced by: fourierswlem 45665 fouriersw 45666 etransclem18 45687 etransclem46 45715

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