iocioodisjd - Mathbox for Steven Nguyen

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Theorem iocioodisjd 43109

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 13383	. . 3 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
5		df-ioo 13382	. . 3 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
6		xrltnle 11282	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵))
7	4, 5, 6	ixxdisj 13393	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)
8	1, 2, 3, 7	syl3anc 1397	1 ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2142 ∩ cin 3903 ∅c0 4285 (class class class)co 7412 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 (,)cioo 13378 (,]cioc 13379

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-xr 11253 df-le 11255 df-ioo 13382 df-ioc 13383

This theorem is used by: readvrec2 43150 readvrec 43151

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