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Theorem xrsex 20960
Description: The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xrsex *𝑠 ∈ V

Proof of Theorem xrsex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xrs 17448 . 2 *𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
2 tpex 7734 . . 3 {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V
3 tpex 7734 . . 3 {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V
42, 3unex 7733 . 2 ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}) ∈ V
51, 4eqeltri 2830 1 *𝑠 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3475  cun 3947  ifcif 4529  {ctp 4633  cop 4635   class class class wbr 5149  cfv 6544  (class class class)co 7409  cmpo 7411  *cxr 11247  cle 11249  -𝑒cxne 13089   +𝑒 cxad 13090   ·e cxmu 13091  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  TopSetcts 17203  lecple 17204  distcds 17206  ordTopcordt 17445  *𝑠cxrs 17446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-tp 4634  df-uni 4910  df-xrs 17448
This theorem is referenced by:  imasdsf1olem  23879  xrslt  32177  xrsmulgzz  32179  xrstos  32180  xrsp0  32182  xrsp1  32183  pnfinf  32329  xrnarchi  32330
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