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Theorem xrsex 21395
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 17547 . 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 7766 . . 3 {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V
3 tpex 7766 . . 3 {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V
42, 3unex 7764 . 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 2837 1 *𝑠 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  cun 3949  ifcif 4525  {ctp 4630  cop 4632   class class class wbr 5143  cfv 6561  (class class class)co 7431  cmpo 7433  *cxr 11294  cle 11296  -𝑒cxne 13151   +𝑒 cxad 13152   ·e cxmu 13153  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  TopSetcts 17303  lecple 17304  distcds 17306  ordTopcordt 17544  *𝑠cxrs 17545
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-tp 4631  df-uni 4908  df-xrs 17547
This theorem is referenced by:  imasdsf1olem  24383  xrslt  33009  xrsmulgzz  33011  xrstos  33012  xrsp0  33014  xrsp1  33015  pnfinf  33190  xrnarchi  33191
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