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Theorem xrnepnf 8450
 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 * A ≠ +∞) ↔ (A A = -∞))

Proof of Theorem xrnepnf
StepHypRef Expression
1 pm5.61 707 . 2 ((((A A = -∞) A = +∞) ¬ A = +∞) ↔ ((A A = -∞) ¬ A = +∞))
2 elxr 8446 . . . 4 (A * ↔ (A A = +∞ A = -∞))
3 df-3or 885 . . . 4 ((A A = +∞ A = -∞) ↔ ((A A = +∞) A = -∞))
4 or32 686 . . . 4 (((A A = +∞) A = -∞) ↔ ((A A = -∞) A = +∞))
52, 3, 43bitri 195 . . 3 (A * ↔ ((A A = -∞) A = +∞))
6 df-ne 2203 . . 3 (A ≠ +∞ ↔ ¬ A = +∞)
75, 6anbi12i 433 . 2 ((A * A ≠ +∞) ↔ (((A A = -∞) A = +∞) ¬ A = +∞))
8 renepnf 6850 . . . . 5 (A ℝ → A ≠ +∞)
9 mnfnepnf 8448 . . . . . 6 -∞ ≠ +∞
10 neeq1 2213 . . . . . 6 (A = -∞ → (A ≠ +∞ ↔ -∞ ≠ +∞))
119, 10mpbiri 157 . . . . 5 (A = -∞ → A ≠ +∞)
128, 11jaoi 635 . . . 4 ((A A = -∞) → A ≠ +∞)
1312neneqd 2221 . . 3 ((A A = -∞) → ¬ A = +∞)
1413pm4.71i 371 . 2 ((A A = -∞) ↔ ((A A = -∞) ¬ A = +∞))
151, 7, 143bitr4i 201 1 ((A * A ≠ +∞) ↔ (A A = -∞))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∨ w3o 883   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  ℝcr 6690  +∞cpnf 6834  -∞cmnf 6835  ℝ*cxr 6836 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-un 4136  ax-cnex 6754  ax-resscn 6755 This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-pnf 6839  df-mnf 6840  df-xr 6841 This theorem is referenced by: (None)
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