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Mirrors > Home > ILE Home > Th. List > ixxssixx | 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 |
---|---|
ixxssixx.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 | ixxssixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
2 | 1 | elmpt2cl 5777 | . . 3 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
3 | simp1 939 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ*) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ*)) |
5 | simpl 107 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
6 | 3simpa 936 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤)) | |
7 | ixx.3 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤)) | |
8 | 7 | expimpd 355 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤) → 𝐴𝑇𝑤)) |
9 | 5, 6, 8 | syl2im 38 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝐴𝑇𝑤)) |
10 | simpr 108 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*) | |
11 | 3simpb 937 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝑤𝑆𝐵)) | |
12 | ixx.4 | . . . . . . . 8 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵)) | |
13 | 12 | ancoms 264 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵)) |
14 | 13 | expimpd 355 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → ((𝑤 ∈ ℝ* ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵)) |
15 | 10, 11, 14 | syl2im 38 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵)) |
16 | 4, 9, 15 | 3jcad 1120 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵))) |
17 | 1 | elixx1 9210 | . . . 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 9210 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵))) |
20 | 16, 17, 19 | 3imtr4d 201 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))) |
21 | 2, 20 | mpcom 36 | . 2 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)) |
22 | 21 | ssriv 3014 | 1 ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 {crab 2357 ⊆ wss 2984 class class class wbr 3811 (class class class)co 5591 ↦ cmpt2 5593 ℝ*cxr 7424 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-pnf 7427 df-mnf 7428 df-xr 7429 |
This theorem is referenced by: ioossicc 9272 icossicc 9273 iocssicc 9274 ioossico 9275 |
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