iocioodisjd - Mathbox for Steven Nguyen

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Theorem iocioodisjd 42315

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 13318	. . 3 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
5		df-ioo 13317	. . 3 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
6		xrltnle 11248	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵))
7	4, 5, 6	ixxdisj 13328	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)
8	1, 2, 3, 7	syl3anc 1373	1 ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∩ cin 3916 ∅c0 4299 (class class class)co 7390 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 (,)cioo 13313 (,]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-ioo 13317 df-ioc 13318

This theorem is referenced by: readvrec2 42356 readvrec 42357

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