iocioodisjd - Mathbox for Steven Nguyen

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Theorem iocioodisjd 43420

Description: Adjacent intervals where the lower interval is right-closed and the upper interval is open are disjoint. (Contributed by SN, 1-Oct-2025.)

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

**Assertion**
Ref	Expression
iocioodisjd	⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)

Proof of Theorem iocioodisjd Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixxdisjd.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		ixxdisjd.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		ixxdisjd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		df-ioc 13481	. . 3 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
5		df-ioo 13480	. . 3 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
6		xrltnle 11376	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵))
7	4, 5, 6	ixxdisj 13491	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)
8	1, 2, 3, 7	syl3anc 1398	1 ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ∩ cin 4089 ∅c0 4278 (class class class)co 7420 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 (,)cioo 13476 (,]cioc 13477

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-xr 11347 df-le 11349 df-ioo 13480 df-ioc 13481

This theorem is used by: readvrec2 43455 readvrec 43456

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