lenelioc - Mathbox for Glauco Siliprandi

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Theorem lenelioc 45541

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 11247	. . . 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 13355	. . 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 1086 ∈ wcel 2109 class class class wbr 5110 (class class class)co 7390 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 (,]cioc 13314

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 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-xr 11219 df-le 11221 df-ioc 13318

This theorem is referenced by: (None)

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