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Theorem renepnfd 10684
 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 10681 . 2 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
31, 2syl 17 1 (𝜑𝐴 ≠ +∞)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 3020  ℝcr 10528  +∞cpnf 10664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454  ax-resscn 10586 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543  df-sn 4564  df-pr 4566  df-uni 4837  df-pnf 10669 This theorem is referenced by:  xaddnepnf  12623  dvfsumrlimge0  24544  dvfsumrlim  24545  dvfsumrlim2  24546  logno1  25134  limsupresico  41848  limsupvaluz2  41886  supcnvlimsup  41888  liminfresico  41919  xlimliminflimsup  42010  smflimsuplem2  42963  smflimsuplem5  42966
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