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Theorem renepnfd 10128
Description: No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
Hypothesis
Ref Expression
rexrd.1 (𝜑𝐴 ∈ ℝ)
Assertion
Ref Expression
renepnfd (𝜑𝐴 ≠ +∞)

Proof of Theorem renepnfd
StepHypRef Expression
1 rexrd.1 . 2 (𝜑𝐴 ∈ ℝ)
2 renepnf 10125 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wne 2823  cr 9973  +∞cpnf 10109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-pnf 10114
This theorem is referenced by:  xaddnepnf  12106  dvfsumrlimge0  23838  dvfsumrlim  23839  dvfsumrlim2  23840  logno1  24427  limsupresico  40250  limsupvaluz2  40288  supcnvlimsup  40290  liminfresico  40321  smflimsuplem2  41348  smflimsuplem5  41351
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