iocioodisjd - Mathbox for Steven Nguyen

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Theorem iocioodisjd 42293

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 13271	. . 3 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
5		df-ioo 13270	. . 3 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
6		xrltnle 11201	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵))
7	4, 5, 6	ixxdisj 13281	. 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 3904 ∅c0 4286 (class class class)co 7353 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 (,)cioo 13266 (,]cioc 13267

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 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-sbc 3745 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-iota 6442 df-fun 6488 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-xr 11172 df-le 11174 df-ioo 13270 df-ioc 13271

This theorem is referenced by: readvrec2 42334 readvrec 42335

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