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Mirrors > Home > MPE Home > Th. List > snunioo | Structured version Visualization version GIF version |
Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
snunioo | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1131 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ*) | |
2 | iccid 12413 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴}) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐴[,]𝐴) = {𝐴}) |
4 | 3 | uneq1d 3909 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) = ({𝐴} ∪ (𝐴(,)𝐵))) |
5 | simp2 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ*) | |
6 | xrleid 12176 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
7 | 1, 6 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐴) |
8 | simp3 1133 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
9 | df-icc 12375 | . . . 4 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
10 | df-ioo 12372 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
11 | xrltnle 10297 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
12 | df-ico 12374 | . . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
13 | xrlelttr 12180 | . . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 𝑤 < 𝐵)) | |
14 | xrltle 12175 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤)) | |
15 | 14 | 3adant1 1125 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤)) |
16 | 15 | adantld 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤)) |
17 | 9, 10, 11, 12, 13, 16 | ixxun 12384 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
18 | 1, 1, 5, 7, 8, 17 | syl32anc 1485 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
19 | 4, 18 | eqtr3d 2796 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∪ cun 3713 {csn 4321 class class class wbr 4804 (class class class)co 6813 ℝ*cxr 10265 < clt 10266 ≤ cle 10267 (,)cioo 12368 [,)cico 12370 [,]cicc 12371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-pre-lttri 10202 ax-pre-lttrn 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-ioo 12372 df-ico 12374 df-icc 12375 |
This theorem is referenced by: prunioo 12494 ioojoin 12496 icombl1 23531 ioombl 23533 tan2h 33714 mbfposadd 33770 itg2addnclem2 33775 ftc1anclem5 33802 iocunico 38298 limciccioolb 40356 fourierdlem32 40859 fourierdlem93 40919 |
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