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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 9669 | . . . 4 ⊢ ℝ* ∈ V | |
2 | 1, 1 | xpex 4662 | . . 3 ⊢ (ℝ* × ℝ*) ∈ V |
3 | 1 | pwex 4115 | . . 3 ⊢ 𝒫 ℝ* ∈ V |
4 | 2, 3 | xpex 4662 | . 2 ⊢ ((ℝ* × ℝ*) × 𝒫 ℝ*) ∈ V |
5 | ixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
6 | 5 | ixxf 9711 | . . 3 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
7 | fssxp 5298 | . . 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 4074 | 1 ⊢ 𝑂 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 {crab 2421 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 class class class wbr 3937 × cxp 4545 ⟶wf 5127 ∈ cmpo 5784 ℝ*cxr 7823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 |
This theorem is referenced by: iooex 9720 |
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