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Theorem xrnepnf 13158
Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xrnepnf ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))

Proof of Theorem xrnepnf
StepHypRef Expression
1 pm5.61 1002 . 2 ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
2 elxr 13156 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3 df-3or 1087 . . . 4 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
4 or32 925 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
52, 3, 43bitri 297 . . 3 (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
6 df-ne 2939 . . 3 (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞)
75, 6anbi12i 628 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞))
8 renepnf 11307 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
9 mnfnepnf 11315 . . . . . 6 -∞ ≠ +∞
10 neeq1 3001 . . . . . 6 (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
119, 10mpbiri 258 . . . . 5 (𝐴 = -∞ → 𝐴 ≠ +∞)
128, 11jaoi 857 . . . 4 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞)
1312neneqd 2943 . . 3 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞)
1413pm4.71i 559 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
151, 7, 143bitr4i 303 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  w3o 1085   = 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:  xaddnepnf  13276  xlt2addrd  32769  xrlexaddrp  45302  xrnpnfmnf  45425
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