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Theorem pnfnre 11201
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
pnfnre +∞ ∉ ℝ

Proof of Theorem pnfnre
StepHypRef Expression
1 df-pnf 11196 . . . 4 +∞ = 𝒫
2 pwuninel 8207 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
31, 2eqneltri 2853 . . 3 ¬ +∞ ∈ ℂ
4 recn 11146 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
53, 4mto 196 . 2 ¬ +∞ ∈ ℝ
65nelir 3049 1 +∞ ∉ ℝ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wnel 3046  𝒫 cpw 4561   cuni 4866  cc 11054  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-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:  pnfnre2  11202  renepnf  11208  ltxrlt  11230  nn0nepnf  12498  xrltnr  13045  pnfnlt  13054  xnn0lenn0nn0  13170  hashclb  14264  hasheq0  14269  pcgcd1  16754  pc2dvds  16756  ramtcl2  16888  odhash3  19363  xrsdsreclblem  20859  pnfnei  22587  iccpnfcnv  24323  i1f0rn  25062
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