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Theorem xrsex 21413
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 17549 . 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 7765 . . 3 {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∈ V
3 tpex 7765 . . 3 {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩} ∈ V
42, 3unex 7763 . 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 2835 1 *𝑠 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  cun 3961  ifcif 4531  {ctp 4635  cop 4637   class class class wbr 5148  cfv 6563  (class class class)co 7431  cmpo 7433  *cxr 11292  cle 11294  -𝑒cxne 13149   +𝑒 cxad 13150   ·e cxmu 13151  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  TopSetcts 17304  lecple 17305  distcds 17307  ordTopcordt 17546  *𝑠cxrs 17547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-tp 4636  df-uni 4913  df-xrs 17549
This theorem is referenced by:  imasdsf1olem  24399  xrslt  32992  xrsmulgzz  32994  xrstos  32995  xrsp0  32997  xrsp1  32998  pnfinf  33173  xrnarchi  33174
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