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Mathbox for Jon Pennant |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iocunico | Structured version Visualization version GIF version |
Description: Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.) |
Ref | Expression |
---|---|
iocunico | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un23 4126 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) | |
2 | unundir 4129 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) | |
3 | uncom 4111 | . . . 4 ⊢ ((𝐵(,)𝐶) ∪ {𝐵}) = ({𝐵} ∪ (𝐵(,)𝐶)) | |
4 | 3 | uneq2i 4118 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) |
5 | 1, 2, 4 | 3eqtrri 2769 | . 2 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) |
6 | simpl1 1191 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ*) | |
7 | simpl2 1192 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ*) | |
8 | simprl 769 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵) | |
9 | ioounsn 13386 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | |
10 | 6, 7, 8, 9 | syl3anc 1371 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
11 | simpl3 1193 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ*) | |
12 | simprr 771 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶) | |
13 | snunioo 13387 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) | |
14 | 7, 11, 12, 13 | syl3anc 1371 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
15 | 10, 14 | uneq12d 4122 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶))) |
16 | ioojoin 13392 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶)) | |
17 | 5, 15, 16 | 3eqtr3a 2800 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∪ cun 3906 {csn 4584 class class class wbr 5103 (class class class)co 7353 ℝ*cxr 11184 < clt 11185 (,)cioo 13256 (,]cioc 13257 [,)cico 13258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-pre-lttri 11121 ax-pre-lttrn 11122 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-ioo 13260 df-ioc 13261 df-ico 13262 df-icc 13263 |
This theorem is referenced by: (None) |
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