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Theorem xnegex 12589
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 12495 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 10686 . . . 4 -∞ ∈ ℝ*
32elexi 3511 . . 3 -∞ ∈ V
4 pnfex 10682 . . . 4 +∞ ∈ V
5 negex 10872 . . . 4 -𝐴 ∈ V
64, 5ifex 4511 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4511 . 2 if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V
81, 7eqeltri 2906 1 -𝑒𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  ifcif 4463  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662  -cneg 10859  -𝑒cxne 12492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-un 7450  ax-cnex 10581
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307  df-fv 6356  df-ov 7148  df-pnf 10665  df-mnf 10666  df-xr 10667  df-neg 10861  df-xneg 12495
This theorem is referenced by:  xrhmeo  23477  supminfxrrnmpt  41623  monoord2xrv  41636  liminfvalxr  41940  liminfpnfuz  41973  xlimpnfxnegmnf2  42015
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