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Mirrors > Home > ILE Home > Th. List > ixxex | 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 8756 | . . . 4 ⊢ ℝ* ∈ V | |
2 | 1, 1 | xpex 4453 | . . 3 ⊢ (ℝ* × ℝ*) ∈ V |
3 | 1 | pwex 3932 | . . 3 ⊢ 𝒫 ℝ* ∈ V |
4 | 2, 3 | xpex 4453 | . 2 ⊢ ((ℝ* × ℝ*) × 𝒫 ℝ*) ∈ V |
5 | ixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
6 | 5 | ixxf 8767 | . . 3 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
7 | fssxp 5058 | . . 3 ⊢ (𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* → 𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*)) | |
8 | 6, 7 | ax-mp 7 | . 2 ⊢ 𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*) |
9 | 4, 8 | ssexi 3895 | 1 ⊢ 𝑂 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 {crab 2310 Vcvv 2557 ⊆ wss 2917 𝒫 cpw 3359 class class class wbr 3764 × cxp 4343 ⟶wf 4898 ↦ cmpt2 5514 ℝ*cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: iooex 8776 |
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