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Theorem xnegex 13136
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 13041 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 11220 . . . 4 -∞ ∈ ℝ*
32elexi 3466 . . 3 -∞ ∈ V
4 pnfex 11216 . . . 4 +∞ ∈ V
5 negex 11407 . . . 4 -𝐴 ∈ V
64, 5ifex 4540 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4540 . 2 if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V
81, 7eqeltri 2830 1 -𝑒𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3447  ifcif 4490  +∞cpnf 11194  -∞cmnf 11195  *cxr 11196  -cneg 11394  -𝑒cxne 13038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-un 7676  ax-cnex 11115
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-uni 4870  df-iota 6452  df-fv 6508  df-ov 7364  df-pnf 11199  df-mnf 11200  df-xr 11201  df-neg 11396  df-xneg 13041
This theorem is referenced by:  xrhmeo  24332  supminfxrrnmpt  43796  monoord2xrv  43809  liminfvalxr  44114  liminfpnfuz  44147  xlimpnfxnegmnf2  44189
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