Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11705 | . . . 4 ⊢ ℝ* ∈ V | |
2 | 1, 1 | xpex 6860 | . . 3 ⊢ (ℝ* × ℝ*) ∈ V |
3 | 1 | pwex 4774 | . . 3 ⊢ 𝒫 ℝ* ∈ V |
4 | 2, 3 | xpex 6860 | . 2 ⊢ ((ℝ* × ℝ*) × 𝒫 ℝ*) ∈ V |
5 | ixx.1 | . . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
6 | 5 | ixxf 12056 | . . 3 ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ* |
7 | fssxp 5973 | . . 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 4731 | 1 ⊢ 𝑂 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ↦ cmpt2 6551 ℝ*cxr 9952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-xr 9957 |
This theorem is referenced by: iooex 12069 isbasisrelowl 32382 relowlpssretop 32388 |
Copyright terms: Public domain | W3C validator |