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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 13388 | . . 3 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
5 | df-ioo 13387 | . . 3 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
6 | xrltnle 11325 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐵)) | |
7 | 4, 5, 6 | ixxdisj 13398 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅) |
8 | 1, 2, 3, 7 | syl3anc 1370 | 1 ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∩ cin 3961 ∅c0 4338 (class class class)co 7430 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 (,)cioo 13383 (,]cioc 13384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-xr 11296 df-le 11298 df-ioo 13387 df-ioc 13388 |
This theorem is referenced by: readvrec2 42369 readvrec 42370 |
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