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Theorem xnegex 12077
Description: A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegex -𝑒𝐴 ∈ V

Proof of Theorem xnegex
StepHypRef Expression
1 df-xneg 11984 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 10134 . . . 4 -∞ ∈ ℝ*
32elexi 3244 . . 3 -∞ ∈ V
4 pnfex 10131 . . . 4 +∞ ∈ V
5 negex 10317 . . . 4 -𝐴 ∈ V
64, 5ifex 4189 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4189 . 2 if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V
81, 7eqeltri 2726 1 -𝑒𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  ifcif 4119  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111  -cneg 10305  -𝑒cxne 11981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-fv 5934  df-ov 6693  df-pnf 10114  df-mnf 10115  df-xr 10116  df-neg 10307  df-xneg 11984
This theorem is referenced by:  xrhmeo  22792  supminfxrrnmpt  40014  monoord2xrv  40027  liminfvalxr  40333
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