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Theorem pnfxr 8793
Description: Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
Assertion
Ref Expression
pnfxr +∞ ∈ ℝ*

Proof of Theorem pnfxr
StepHypRef Expression
1 ssun2 3135 . . 3 {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
2 df-pnf 7121 . . . . 5 +∞ = 𝒫
3 cnex 7063 . . . . . . 7 ℂ ∈ V
43uniex 4202 . . . . . 6 ℂ ∈ V
54pwex 3960 . . . . 5 𝒫 ℂ ∈ V
62, 5eqeltri 2126 . . . 4 +∞ ∈ V
76prid1 3504 . . 3 +∞ ∈ {+∞, -∞}
81, 7sselii 2970 . 2 +∞ ∈ (ℝ ∪ {+∞, -∞})
9 df-xr 7123 . 2 * = (ℝ ∪ {+∞, -∞})
108, 9eleqtrri 2129 1 +∞ ∈ ℝ*
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  cun 2943  𝒫 cpw 3387  {cpr 3404   cuni 3608  cc 6945  cr 6946  +∞cpnf 7116  -∞cmnf 7117  *cxr 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-un 4198  ax-cnex 7033
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-pnf 7121  df-xr 7123
This theorem is referenced by:  pnfex  8794  pnfnemnf  8798  xrltnr  8802  ltpnf  8803  mnfltpnf  8807  pnfnlt  8809  pnfge  8811  xrlttri3  8819  nltpnft  8831  xrrebnd  8833  xrre  8834  xrre2  8835  xnegcl  8846  xrex  8857  elioc2  8906  elico2  8907  elicc2  8908  ioomax  8918  iccmax  8919  ioopos  8920  elioopnf  8937  elicopnf  8939  unirnioo  8943  elxrge0  8948
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