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Theorem renepnfd 11211
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 11208 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2940  cr 11055  +∞cpnf 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-pr 5385  ax-un 7673  ax-resscn 11113
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-nel 3047  df-rab 3407  df-v 3446  df-un 3916  df-in 3918  df-ss 3928  df-pw 4563  df-sn 4588  df-pr 4590  df-uni 4867  df-pnf 11196
This theorem is referenced by:  xaddnepnf  13162  dvfsumrlimge0  25410  dvfsumrlim  25411  dvfsumrlim2  25412  logno1  26007  limsupresico  44027  limsupvaluz2  44065  supcnvlimsup  44067  liminfresico  44098  xlimliminflimsup  44189  smflimsuplem2  45148  smflimsuplem5  45151
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