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Theorem mnfxr 8795
Description: Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
mnfxr -∞ ∈ ℝ*

Proof of Theorem mnfxr
StepHypRef Expression
1 df-mnf 7122 . . . . 5 -∞ = 𝒫 +∞
2 pnfex 8794 . . . . . 6 +∞ ∈ V
32pwex 3960 . . . . 5 𝒫 +∞ ∈ V
41, 3eqeltri 2126 . . . 4 -∞ ∈ V
54prid2 3505 . . 3 -∞ ∈ {+∞, -∞}
6 elun2 3139 . . 3 (-∞ ∈ {+∞, -∞} → -∞ ∈ (ℝ ∪ {+∞, -∞}))
75, 6ax-mp 7 . 2 -∞ ∈ (ℝ ∪ {+∞, -∞})
8 df-xr 7123 . 2 * = (ℝ ∪ {+∞, -∞})
97, 8eleqtrri 2129 1 -∞ ∈ ℝ*
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  cun 2943  𝒫 cpw 3387  {cpr 3404  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-mnf 7122  df-xr 7123
This theorem is referenced by:  elxr  8797  xrltnr  8802  mnflt  8805  mnfltpnf  8807  nltmnf  8810  mnfle  8814  xrltnsym  8815  xrlttri3  8819  ngtmnft  8832  xrrebnd  8833  xrre2  8835  xrre3  8836  ge0gtmnf  8837  xnegcl  8846  xltnegi  8849  xrex  8857  elioc2  8906  elico2  8907  elicc2  8908  ioomax  8918  iccmax  8919  elioomnf  8938  unirnioo  8943
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