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Theorem xrnpnfmnf 45425
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 511 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ +∞))
4 xrnepnf 13158 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
53, 4sylib 218 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6 xrnpnfmnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 851 . 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 395  wo 847   = wceq 1537  wcel 2106  wne 2938  cr 11152  +∞cpnf 11290  -∞cmnf 11291  *cxr 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-pnf 11295  df-mnf 11296  df-xr 11297
This theorem is referenced by:  xlimliminflimsup  45818
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