iocioodisjd - Mathbox for Steven Nguyen

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Theorem iocioodisjd 43182

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 13405	. . 3 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
5		df-ioo 13404	. . 3 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
6		xrltnle 11303	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵))
7	4, 5, 6	ixxdisj 13415	. 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 1570 ∈ wcel 2145 ∩ cin 3901 ∅c0 4282 (class class class)co 7416 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 (,)cioo 13400 (,]cioc 13401

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 2215 ax-ext 2734 ax-sep 5255 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-xr 11274 df-le 11276 df-ioo 13404 df-ioc 13405

This theorem is used by: readvrec2 43223 readvrec 43224

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