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Theorem xrnemnf 8469
Description: An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xrnemnf ((A * A ≠ -∞) ↔ (A A = +∞))

Proof of Theorem xrnemnf
StepHypRef Expression
1 pm5.61 707 . 2 ((((A A = +∞) A = -∞) ¬ A = -∞) ↔ ((A A = +∞) ¬ A = -∞))
2 elxr 8466 . . . 4 (A * ↔ (A A = +∞ A = -∞))
3 df-3or 885 . . . 4 ((A A = +∞ A = -∞) ↔ ((A A = +∞) A = -∞))
42, 3bitri 173 . . 3 (A * ↔ ((A A = +∞) A = -∞))
5 df-ne 2203 . . 3 (A ≠ -∞ ↔ ¬ A = -∞)
64, 5anbi12i 433 . 2 ((A * A ≠ -∞) ↔ (((A A = +∞) A = -∞) ¬ A = -∞))
7 renemnf 6871 . . . . 5 (A ℝ → A ≠ -∞)
8 pnfnemnf 8467 . . . . . 6 +∞ ≠ -∞
9 neeq1 2213 . . . . . 6 (A = +∞ → (A ≠ -∞ ↔ +∞ ≠ -∞))
108, 9mpbiri 157 . . . . 5 (A = +∞ → A ≠ -∞)
117, 10jaoi 635 . . . 4 ((A A = +∞) → A ≠ -∞)
1211neneqd 2221 . . 3 ((A A = +∞) → ¬ A = -∞)
1312pm4.71i 371 . 2 ((A A = +∞) ↔ ((A A = +∞) ¬ A = -∞))
141, 6, 133bitr4i 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  cr 6710  +∞cpnf 6854  -∞cmnf 6855  *cxr 6856
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-setind 4220  ax-cnex 6774  ax-resscn 6775
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-pnf 6859  df-mnf 6860  df-xr 6861
This theorem is referenced by: (None)
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