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| Mirrors > Home > MPE Home > Th. List > ioojoin | Structured version Visualization version GIF version | ||
| Description: Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.) |
| Ref | Expression |
|---|---|
| ioojoin | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unass 4172 | . 2 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = ((𝐴(,)𝐵) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) | |
| 2 | snunioo 13518 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) | |
| 3 | 2 | 3expa 1119 | . . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
| 4 | 3 | 3adantl1 1167 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
| 5 | 4 | adantrl 716 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
| 6 | 5 | uneq2d 4168 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶))) |
| 7 | df-ioo 13391 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 8 | df-ico 13393 | . . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 9 | xrlenlt 11326 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵)) | |
| 10 | xrlttr 13182 | . . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤 < 𝐵 ∧ 𝐵 < 𝐶) → 𝑤 < 𝐶)) | |
| 11 | xrltletr 13199 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤) → 𝐴 < 𝑤)) | |
| 12 | 7, 8, 9, 7, 10, 11 | ixxun 13403 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)) |
| 13 | 6, 12 | eqtrd 2777 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = (𝐴(,)𝐶)) |
| 14 | 1, 13 | eqtrid 2789 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 [,)cico 13389 |
| 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-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-ico 13393 df-icc 13394 |
| This theorem is referenced by: reconnlem1 24848 itgsplitioo 25873 lhop 26055 iocunico 43223 |
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