MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  renepnfd Structured version   Visualization version   GIF version

Theorem renepnfd 11270
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 11267 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wne 2939  cr 11112  +∞cpnf 11250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pr 5427  ax-un 7728  ax-resscn 11170
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-rab 3432  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909  df-pnf 11255
This theorem is referenced by:  xaddnepnf  13221  dvfsumrlimge0  25783  dvfsumrlim  25784  dvfsumrlim2  25785  logno1  26381  limsupresico  44715  limsupvaluz2  44753  supcnvlimsup  44755  liminfresico  44786  xlimliminflimsup  44877  smflimsuplem2  45836  smflimsuplem5  45839
  Copyright terms: Public domain W3C validator