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Mirrors > Home > MPE Home > Th. List > snunico | Structured version Visualization version GIF version |
Description: The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
snunico | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1131 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) | |
2 | iccid 12405 | . . . 4 ⊢ (𝐵 ∈ ℝ* → (𝐵[,]𝐵) = {𝐵}) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐵[,]𝐵) = {𝐵}) |
4 | 3 | uneq2d 3902 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = ((𝐴[,)𝐵) ∪ {𝐵})) |
5 | simp1 1130 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) | |
6 | simp3 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
7 | xrleid 12168 | . . . 4 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵) |
9 | df-ico 12366 | . . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | df-icc 12367 | . . . 4 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
11 | xrlenlt 10287 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵)) | |
12 | xrltle 12167 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
13 | 12 | 3adant3 1126 | . . . . 5 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) |
14 | 13 | adantrd 485 | . . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵) → 𝑤 ≤ 𝐵)) |
15 | xrletr 12174 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝑤) → 𝐴 ≤ 𝑤)) | |
16 | 9, 10, 11, 10, 14, 15 | ixxun 12376 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴[,]𝐵)) |
17 | 5, 1, 1, 6, 8, 16 | syl32anc 1481 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴[,]𝐵)) |
18 | 4, 17 | eqtr3d 2788 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 ∪ cun 3705 {csn 4313 class class class wbr 4796 (class class class)co 6805 ℝ*cxr 10257 < clt 10258 ≤ cle 10259 [,)cico 12362 [,]cicc 12363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-pre-lttri 10194 ax-pre-lttrn 10195 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-ico 12366 df-icc 12367 |
This theorem is referenced by: prunioo 12486 iccpnfcnv 22936 iccpnfhmeo 22937 xrge0iifcnv 30280 |
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