lenelioc - Mathbox for Glauco Siliprandi

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Theorem lenelioc 45549

Description: A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

**Hypotheses**
Ref	Expression
lenelioc.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
lenelioc.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
lenelioc.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
lenelioc.4	⊢ (𝜑 → 𝐶 ≤ 𝐴)

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

**Proof of Theorem lenelioc**
Step	Hyp	Ref	Expression
1		lenelioc.4	. . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴)
2		lenelioc.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		lenelioc.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4	2, 3	xrlenltd 11327	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶))
5	1, 4	mpbid 232	. . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶)
6	5	intn3an2d 1482	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
7		lenelioc.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		elioc1 13429	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
9	3, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	6, 9	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 (,]cioc 13388

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-le 11301 df-ioc 13392

This theorem is referenced by: (None)

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