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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iocioodisjd | Structured version Visualization version GIF version | ||
| Description: Adjacent intervals where the lower interval is right-closed and the upper interval is open are disjoint. (Contributed by SN, 1-Oct-2025.) |
| Ref | Expression |
|---|---|
| ixxdisjd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| ixxdisjd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| ixxdisjd.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
| Ref | Expression |
|---|---|
| iocioodisjd | ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅) |
| 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 13392 | . . 3 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
| 5 | df-ioo 13391 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 6 | xrltnle 11328 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵)) | |
| 7 | 4, 5, 6 | ixxdisj 13402 | . 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 2108 ∩ cin 3950 ∅c0 4333 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 (,]cioc 13388 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-le 11301 df-ioo 13391 df-ioc 13392 |
| This theorem is referenced by: readvrec2 42391 readvrec 42392 |
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