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Theorem xrnepnf 8698
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 pm5.61 708 . 2 ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
2 elxr 8694 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3 df-3or 886 . . . 4 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
4 or32 687 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
52, 3, 43bitri 195 . . 3 (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
6 df-ne 2206 . . 3 (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞)
75, 6anbi12i 433 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞))
8 renepnf 7071 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
9 mnfnepnf 8696 . . . . . 6 -∞ ≠ +∞
10 neeq1 2218 . . . . . 6 (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
119, 10mpbiri 157 . . . . 5 (𝐴 = -∞ → 𝐴 ≠ +∞)
128, 11jaoi 636 . . . 4 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞)
1312neneqd 2226 . . 3 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞)
1413pm4.71i 371 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
151, 7, 143bitr4i 201 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wb 98  wo 629  w3o 884   = wceq 1243  wcel 1393  wne 2204  cr 6886  +∞cpnf 7055  -∞cmnf 7056  *cxr 7057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-un 4170  ax-cnex 6973  ax-resscn 6974
This theorem depends on definitions:  df-bi 110  df-3or 886  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7060  df-mnf 7061  df-xr 7062
This theorem is referenced by: (None)
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