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Theorem xrsex 19680
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 16083 . 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 6910 . . 3 {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V
3 tpex 6910 . . 3 {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V
42, 3unex 6909 . 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 2694 1 *𝑠 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  Vcvv 3186  cun 3553  ifcif 4058  {ctp 4152  cop 4154   class class class wbr 4613  cfv 5847  (class class class)co 6604  cmpt2 6606  *cxr 10017  cle 10019  -𝑒cxne 11887   +𝑒 cxad 11888   ·e cxmu 11889  ndxcnx 15778  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  TopSetcts 15868  lecple 15869  distcds 15871  ordTopcordt 16080  *𝑠cxrs 16081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151  df-tp 4153  df-uni 4403  df-xrs 16083
This theorem is referenced by:  imasdsf1olem  22088  xrslt  29461  xrsmulgzz  29463  xrstos  29464  xrsp0  29466  xrsp1  29467  pnfinf  29522  xrnarchi  29523
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