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Theorem xrsex 20894
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 17430 . 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 7717 . . 3 {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V
3 tpex 7717 . . 3 {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V
42, 3unex 7716 . 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 2828 1 *𝑠 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3473  cun 3942  ifcif 4522  {ctp 4626  cop 4628   class class class wbr 5141  cfv 6532  (class class class)co 7393  cmpo 7395  *cxr 11229  cle 11231  -𝑒cxne 13071   +𝑒 cxad 13072   ·e cxmu 13073  ndxcnx 17108  Basecbs 17126  +gcplusg 17179  .rcmulr 17180  TopSetcts 17185  lecple 17186  distcds 17188  ordTopcordt 17427  *𝑠cxrs 17428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4623  df-pr 4625  df-tp 4627  df-uni 4902  df-xrs 17430
This theorem is referenced by:  imasdsf1olem  23808  xrslt  32048  xrsmulgzz  32050  xrstos  32051  xrsp0  32053  xrsp1  32054  pnfinf  32200  xrnarchi  32201
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