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Mirrors > Home > ILE Home > Th. List > ixxss1 | GIF version |
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ixxssixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
ixxss1.2 | ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) |
ixxss1.3 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
Ref | Expression |
---|---|
ixxss1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss1.2 | . . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) | |
2 | 1 | elixx3g 9467 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))) |
3 | 2 | simplbi 269 | . . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
4 | 3 | adantl 272 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
5 | 4 | simp3d 960 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ*) |
6 | simplr 498 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵) | |
7 | 2 | simprbi 270 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
8 | 7 | adantl 272 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
9 | 8 | simpld 111 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤) |
10 | simpll 497 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ*) | |
11 | 4 | simp1d 958 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ*) |
12 | ixxss1.3 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) | |
13 | 10, 11, 5, 12 | syl3anc 1181 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
14 | 6, 9, 13 | mp2and 425 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤) |
15 | 8 | simprd 113 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶) |
16 | 4 | simp2d 959 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ*) |
17 | ixxssixx.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
18 | 17 | elixx1 9463 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
19 | 10, 16, 18 | syl2anc 404 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
20 | 5, 14, 15, 19 | mpbir3and 1129 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶)) |
21 | 20 | ex 114 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶))) |
22 | 21 | ssrdv 3045 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 {crab 2374 ⊆ wss 3013 class class class wbr 3867 (class class class)co 5690 ↦ cmpt2 5692 ℝ*cxr 7618 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 |
This theorem is referenced by: iooss1 9482 |
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