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Mirrors > Home > MPE Home > Th. List > ixxss1 | 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)}) |
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 12754 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))) |
3 | 2 | simplbi 500 | . . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
4 | 3 | adantl 484 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
5 | 4 | simp3d 1140 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ*) |
6 | simplr 767 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵) | |
7 | 2 | simprbi 499 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
8 | 7 | adantl 484 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
9 | 8 | simpld 497 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤) |
10 | simpll 765 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ*) | |
11 | 4 | simp1d 1138 | . . . . . 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 1367 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
14 | 6, 9, 13 | mp2and 697 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤) |
15 | 8 | simprd 498 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶) |
16 | 4 | simp2d 1139 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ*) |
17 | ixx.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
18 | 17 | elixx1 12750 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
19 | 10, 16, 18 | syl2anc 586 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
20 | 5, 14, 15, 19 | mpbir3and 1338 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶)) |
21 | 20 | ex 415 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶))) |
22 | 21 | ssrdv 3976 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 ⊆ wss 3939 class class class wbr 5069 (class class class)co 7159 ∈ cmpo 7161 ℝ*cxr 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-xr 10682 |
This theorem is referenced by: iooss1 12776 limsupgord 14832 pnfnei 21831 dvfsumrlimge0 24630 dvfsumrlim2 24632 tanord1 25124 rlimcnp 25546 rlimcnp2 25547 dchrisum0lem2a 26096 pntleml 26190 pnt 26193 liminfgord 42041 |
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