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Theorem renepnfd 11269
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 11266 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  wne 2938  cr 11111  +∞cpnf 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-pr 5426  ax-un 7727  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-nel 3045  df-rab 3431  df-v 3474  df-un 3952  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-pnf 11254
This theorem is referenced by:  xaddnepnf  13220  dvfsumrlimge0  25782  dvfsumrlim  25783  dvfsumrlim2  25784  logno1  26380  limsupresico  44714  limsupvaluz2  44752  supcnvlimsup  44754  liminfresico  44785  xlimliminflimsup  44876  smflimsuplem2  45835  smflimsuplem5  45838
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