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Theorem renepnf 6870
 Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renepnf (A ℝ → A ≠ +∞)

Proof of Theorem renepnf
StepHypRef Expression
1 pnfnre 6864 . . . 4 +∞ ∉ ℝ
21neli 2293 . . 3 ¬ +∞
3 eleq1 2097 . . 3 (A = +∞ → (A ℝ ↔ +∞ ℝ))
42, 3mtbiri 599 . 2 (A = +∞ → ¬ A ℝ)
54necon2ai 2253 1 (A ℝ → A ≠ +∞)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  ℝcr 6710  +∞cpnf 6854 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-bndl 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-un 4136  ax-cnex 6774  ax-resscn 6775 This theorem depends on definitions:  df-bi 110  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-in 2918  df-ss 2925  df-pw 3353  df-uni 3572  df-pnf 6859 This theorem is referenced by:  renepnfd  6873  renfdisj  6876  ltxrlt  6882  xrnepnf  8470  xrlttri3  8488  nltpnft  8500  xrrebnd  8502  rexneg  8513
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