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| Mirrors > Home > MPE Home > Th. List > ixxex | Structured version Visualization version GIF version | ||
| Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
| Ref | Expression |
|---|---|
| ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
| Ref | Expression |
|---|---|
| ixxex | ⊢ 𝑂 ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrex 13029 | . . . 4 ⊢ ℝ* ∈ V | |
| 2 | 1, 1 | xpex 7773 | . . 3 ⊢ (ℝ* × ℝ*) ∈ V |
| 3 | 1 | pwex 5380 | . . 3 ⊢ 𝒫 ℝ* ∈ V |
| 4 | 2, 3 | xpex 7773 | . 2 ⊢ ((ℝ* × ℝ*) × 𝒫 ℝ*) ∈ V |
| 5 | ixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
| 6 | 5 | ixxf 13397 | . . 3 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
| 7 | fssxp 6763 | . . 3 ⊢ (𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* → 𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*)) | |
| 8 | 6, 7 | ax-mp 5 | . 2 ⊢ 𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*) |
| 9 | 4, 8 | ssexi 5322 | 1 ⊢ 𝑂 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 × cxp 5683 ⟶wf 6557 ∈ cmpo 7433 ℝ*cxr 11294 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-xr 11299 |
| This theorem is referenced by: iooex 13410 isbasisrelowl 37359 relowlpssretop 37365 |
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