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Theorem xrnpnfmnf 43717
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 13040 . . 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 2944  cr 11051  +∞cpnf 11187  -∞cmnf 11188  *cxr 11189
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 2708  ax-sep 5257  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-rab 3409  df-v 3448  df-un 3916  df-in 3918  df-ss 3928  df-pw 4563  df-sn 4588  df-pr 4590  df-uni 4867  df-pnf 11192  df-mnf 11193  df-xr 11194
This theorem is referenced by:  xlimliminflimsup  44110
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