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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 4129 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) | |
2 | unundir 4132 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ (𝐵(,)𝐶)) ∪ {𝐵}) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) | |
3 | uncom 4114 | . . . 4 ⊢ ((𝐵(,)𝐶) ∪ {𝐵}) = ({𝐵} ∪ (𝐵(,)𝐶)) | |
4 | 3 | uneq2i 4121 | . . 3 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ((𝐵(,)𝐶) ∪ {𝐵})) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) |
5 | 1, 2, 4 | 3eqtrri 2766 | . 2 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) |
6 | simpl1 1192 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ*) | |
7 | simpl2 1193 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ*) | |
8 | simprl 770 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵) | |
9 | ioounsn 13400 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | |
10 | 6, 7, 8, 9 | syl3anc 1372 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
11 | simpl3 1194 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ*) | |
12 | simprr 772 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶) | |
13 | snunioo 13401 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 < 𝐶) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) | |
14 | 7, 11, 12, 13 | syl3anc 1372 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∪ (𝐵(,)𝐶)) = (𝐵[,)𝐶)) |
15 | 10, 14 | uneq12d 4125 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ ({𝐵} ∪ (𝐵(,)𝐶))) = ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶))) |
16 | ioojoin 13406 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶)) | |
17 | 5, 15, 16 | 3eqtr3a 2797 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {csn 4587 class class class wbr 5106 (class class class)co 7358 ℝ*cxr 11193 < clt 11194 (,)cioo 13270 (,]cioc 13271 [,)cico 13272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 |
This theorem is referenced by: (None) |
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