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Theorem renepnf 7438
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 7432 . . . 4 +∞ ∉ ℝ
21neli 2346 . . 3 ¬ +∞ ∈ ℝ
3 eleq1 2145 . . 3 (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ))
42, 3mtbiri 633 . 2 (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ)
54necon2ai 2303 1 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  wne 2249  cr 7252  +∞cpnf 7422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-un 4224  ax-cnex 7339  ax-resscn 7340
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-rex 2359  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-uni 3628  df-pnf 7427
This theorem is referenced by:  renepnfd  7441  renfdisj  7449  ltxrlt  7455  xrnepnf  9144  xrlttri3  9162  nltpnft  9174  xrrebnd  9176  rexneg  9187
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