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Theorem xnegex 11989
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 11897 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2 mnfxr 10047 . . . 4 -∞ ∈ ℝ*
32elexi 3202 . . 3 -∞ ∈ V
4 pnfex 10044 . . . 4 +∞ ∈ V
5 negex 10230 . . . 4 -𝐴 ∈ V
64, 5ifex 4133 . . 3 if(𝐴 = -∞, +∞, -𝐴) ∈ V
73, 6ifex 4133 . 2 if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) ∈ V
81, 7eqeltri 2694 1 -𝑒𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3189  ifcif 4063  +∞cpnf 10022  -∞cmnf 10023  *cxr 10024  -cneg 10218  -𝑒cxne 11894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-un 6909  ax-cnex 9943
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5815  df-fv 5860  df-ov 6613  df-pnf 10027  df-mnf 10028  df-xr 10029  df-neg 10220  df-xneg 11897
This theorem is referenced by:  xrhmeo  22664
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