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Theorem xrnpnfmnf 42282
 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 515 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ +∞))
4 xrnepnf 12521 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
53, 4sylib 221 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6 xrnpnfmnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 848 . 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 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ℝcr 10543  +∞cpnf 10679  -∞cmnf 10680  ℝ*cxr 10681 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4805  df-pnf 10684  df-mnf 10685  df-xr 10686 This theorem is referenced by:  xlimliminflimsup  42672
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