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Theorem xrnpnfmnf 44185
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 513 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ +∞))
4 xrnepnf 13098 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
53, 4sylib 217 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6 xrnpnfmnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 850 . 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 397  wo 846   = wceq 1542  wcel 2107  wne 2941  cr 11109  +∞cpnf 11245  -∞cmnf 11246  *cxr 11247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-nel 3048  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-pnf 11250  df-mnf 11251  df-xr 11252
This theorem is referenced by:  xlimliminflimsup  44578
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