lenelioc - Mathbox for Glauco Siliprandi

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Theorem lenelioc 46366

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 11299	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶))
5	1, 4	mpbid 235	. . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶)
6	5	intn3an2d 1511	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
7		lenelioc.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		elioc1 13440	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
9	3, 7, 8	syl2anc 596	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	6, 9	mtbird 328	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ w3a 1103 ∈ wcel 2145 class class class wbr 5103 (class class class)co 7413 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 (,]cioc 13399

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-xr 11271 df-le 11273 df-ioc 13403

This theorem is used by: (None)

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