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Mirrors > Home > MPE Home > Th. List > ixxss2 | Structured version Visualization version 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 |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
ixxss2.2 | ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)}) |
ixxss2.3 | ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶)) |
Ref | Expression |
---|---|
ixxss2 | ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss2.2 | . . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)}) | |
2 | 1 | elixx3g 12188 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))) |
3 | 2 | simplbi 476 | . . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
4 | 3 | adantl 482 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
5 | 4 | simp3d 1075 | . . . 4 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ ℝ*) |
6 | 2 | simprbi 480 | . . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)) |
7 | 6 | adantl 482 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)) |
8 | 7 | simpld 475 | . . . 4 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴𝑅𝑤) |
9 | 7 | simprd 479 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑇𝐵) |
10 | simplr 792 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵𝑊𝐶) | |
11 | 4 | simp2d 1074 | . . . . . 6 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵 ∈ ℝ*) |
12 | simpll 790 | . . . . . 6 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐶 ∈ ℝ*) | |
13 | ixxss2.3 | . . . . . 6 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶)) | |
14 | 5, 11, 12, 13 | syl3anc 1326 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶)) |
15 | 9, 10, 14 | mp2and 715 | . . . 4 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑆𝐶) |
16 | 4 | simp1d 1073 | . . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴 ∈ ℝ*) |
17 | ixx.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
18 | 17 | elixx1 12184 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
19 | 16, 12, 18 | syl2anc 693 | . . . 4 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
20 | 5, 8, 15, 19 | mpbir3and 1245 | . . 3 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ (𝐴𝑂𝐶)) |
21 | 20 | ex 450 | . 2 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) → (𝑤 ∈ (𝐴𝑃𝐵) → 𝑤 ∈ (𝐴𝑂𝐶))) |
22 | 21 | ssrdv 3609 | 1 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 ⊆ wss 3574 class class class wbr 4653 (class class class)co 6650 ↦ cmpt2 6652 ℝ*cxr 10073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-xr 10078 |
This theorem is referenced by: iooss2 12211 leordtval2 21016 mnfnei 21025 psercnlem2 24178 tanord1 24283 |
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