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Theorem xnegex 12253
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 12158 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 10378 . . . 4 -∞ ∈ ℝ*
32elexi 3406 . . 3 -∞ ∈ V
4 pnfex 10375 . . . 4 +∞ ∈ V
5 negex 10561 . . . 4 -𝐴 ∈ V
64, 5ifex 4324 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4324 . 2 if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V
81, 7eqeltri 2880 1 -𝑒𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2158  Vcvv 3390  ifcif 4276  +∞cpnf 10353  -∞cmnf 10354  *cxr 10355  -cneg 10549  -𝑒cxne 12155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-un 7176  ax-cnex 10274
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rex 3101  df-v 3392  df-sbc 3631  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-uni 4627  df-iota 6061  df-fv 6106  df-ov 6874  df-pnf 10358  df-mnf 10359  df-xr 10360  df-neg 10551  df-xneg 12158
This theorem is referenced by:  xrhmeo  22954  supminfxrrnmpt  40177  monoord2xrv  40190  liminfvalxr  40492
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