MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixxex Structured version   Visualization version   GIF version

Theorem ixxex 13398
Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
ixx.1 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
Assertion
Ref Expression
ixxex 𝑂 ∈ V
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝑆,𝑦,𝑧
Allowed substitution hints:   𝑂(𝑥,𝑦,𝑧)

Proof of Theorem ixxex
StepHypRef Expression
1 xrex 13029 . . . 4 * ∈ V
21, 1xpex 7773 . . 3 (ℝ* × ℝ*) ∈ V
31pwex 5380 . . 3 𝒫 ℝ* ∈ V
42, 3xpex 7773 . 2 ((ℝ* × ℝ*) × 𝒫 ℝ*) ∈ V
5 ixx.1 . . . 4 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})
65ixxf 13397 . . 3 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
7 fssxp 6763 . . 3 (𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*))
86, 7ax-mp 5 . 2 𝑂 ⊆ ((ℝ* × ℝ*) × 𝒫 ℝ*)
94, 8ssexi 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  *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
  Copyright terms: Public domain W3C validator