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Theorem renepnfd 11310
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 11307 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  cr 11152  +∞cpnf 11290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-pnf 11295
This theorem is referenced by:  xaddnepnf  13276  dvfsumrlimge0  26086  dvfsumrlim  26087  dvfsumrlim2  26088  logno1  26693  limsupresico  45656  limsupvaluz2  45694  supcnvlimsup  45696  liminfresico  45727  xlimliminflimsup  45818  smflimsuplem2  46777  smflimsuplem5  46780
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