![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ixxssixx | Structured version Visualization version GIF version |
Description: An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
ixx.2 | ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)}) |
ixx.3 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤)) |
ixx.4 | ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵)) |
Ref | Expression |
---|---|
ixxssixx | ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
2 | 1 | elmpocl 7367 | . . 3 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
3 | simp1 1133 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ*) | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ*)) |
5 | simpl 486 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
6 | 3simpa 1145 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤)) | |
7 | ixx.3 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤)) | |
8 | 7 | expimpd 457 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤) → 𝐴𝑇𝑤)) |
9 | 5, 6, 8 | syl2im 40 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝐴𝑇𝑤)) |
10 | simpr 488 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*) | |
11 | 3simpb 1146 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝑤𝑆𝐵)) | |
12 | ixx.4 | . . . . . . . 8 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵)) | |
13 | 12 | ancoms 462 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵)) |
14 | 13 | expimpd 457 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → ((𝑤 ∈ ℝ* ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵)) |
15 | 10, 11, 14 | syl2im 40 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵)) |
16 | 4, 9, 15 | 3jcad 1126 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵))) |
17 | 1 | elixx1 12735 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵))) |
18 | ixx.2 | . . . . 5 ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)}) | |
19 | 18 | elixx1 12735 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵))) |
20 | 16, 17, 19 | 3imtr4d 297 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))) |
21 | 2, 20 | mpcom 38 | . 2 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)) |
22 | 21 | ssriv 3919 | 1 ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 class class class wbr 5030 (class class class)co 7135 ∈ cmpo 7137 ℝ*cxr 10663 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-xr 10668 |
This theorem is referenced by: ioossicc 12811 icossicc 12814 iocssicc 12815 ioossico 12816 dvloglem 25239 ioossioc 42129 |
Copyright terms: Public domain | W3C validator |