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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 4068 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) | |
2 | unundir 4071 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) | |
3 | uncom 4053 | . . . 4 ⊢ ((𝐵(,)𝐶) ∪ {𝐵}) = ({𝐵} ∪ (𝐵(,)𝐶)) | |
4 | 3 | uneq2i 4060 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) |
5 | 1, 2, 4 | 3eqtrri 2764 | . 2 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) |
6 | simpl1 1193 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ*) | |
7 | simpl2 1194 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ*) | |
8 | simprl 771 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵) | |
9 | ioounsn 13030 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | |
10 | 6, 7, 8, 9 | syl3anc 1373 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
11 | simpl3 1195 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ*) | |
12 | simprr 773 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶) | |
13 | snunioo 13031 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) | |
14 | 7, 11, 12, 13 | syl3anc 1373 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
15 | 10, 14 | uneq12d 4064 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶))) |
16 | ioojoin 13036 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶)) | |
17 | 5, 15, 16 | 3eqtr3a 2795 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 {csn 4527 class class class wbr 5039 (class class class)co 7191 ℝ*cxr 10831 < clt 10832 (,)cioo 12900 (,]cioc 12901 [,)cico 12902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 |
This theorem is referenced by: (None) |
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