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Theorem xnegex 13187
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 13092 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 11271 . . . 4 -∞ ∈ ℝ*
32elexi 3494 . . 3 -∞ ∈ V
4 pnfex 11267 . . . 4 +∞ ∈ V
5 negex 11458 . . . 4 -𝐴 ∈ V
64, 5ifex 4579 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4579 . 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 3475  ifcif 4529  +∞cpnf 11245  -∞cmnf 11246  *cxr 11247  -cneg 11445  -𝑒cxne 13089
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 5300  ax-nul 5307  ax-pow 5364  ax-un 7725  ax-cnex 11166
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 2942  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496  df-fv 6552  df-ov 7412  df-pnf 11250  df-mnf 11251  df-xr 11252  df-neg 11447  df-xneg 13092
This theorem is referenced by:  xrhmeo  24462  supminfxrrnmpt  44181  monoord2xrv  44194  liminfvalxr  44499  liminfpnfuz  44532  xlimpnfxnegmnf2  44574
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