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Theorem xrnpnfmnf 44264
Description: An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
xrnpnfmnf.1 (𝜑𝐴 ∈ ℝ*)
xrnpnfmnf.2 (𝜑 → ¬ 𝐴 ∈ ℝ)
xrnpnfmnf.3 (𝜑𝐴 ≠ +∞)
Assertion
Ref Expression
xrnpnfmnf (𝜑𝐴 = -∞)

Proof of Theorem xrnpnfmnf
StepHypRef Expression
1 xrnpnfmnf.1 . . . 4 (𝜑𝐴 ∈ ℝ*)
2 xrnpnfmnf.3 . . . 4 (𝜑𝐴 ≠ +∞)
31, 2jca 512 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ +∞))
4 xrnepnf 13100 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
53, 4sylib 217 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6 xrnpnfmnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 849 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞))
85, 6, 7sylc 65 1 (𝜑𝐴 = -∞)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2940  cr 11111  +∞cpnf 11247  -∞cmnf 11248  *cxr 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-rab 3433  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909  df-pnf 11252  df-mnf 11253  df-xr 11254
This theorem is referenced by:  xlimliminflimsup  44657
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