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Theorem pnfnre 8030
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 cnex 7966 . . . . . 6 ℂ ∈ V
21uniex 4455 . . . . 5 ℂ ∈ V
3 pwuninel2 6308 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 5 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 8025 . . . . 5 +∞ = 𝒫
65eleq1i 2255 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 672 . . 3 ¬ +∞ ∈ ℂ
8 recn 7975 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 663 . 2 ¬ +∞ ∈ ℝ
109nelir 2458 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2160  wnel 2455  Vcvv 2752  𝒫 cpw 3590   cuni 3824  cc 7840  cr 7841  +∞cpnf 8020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451  ax-cnex 7933  ax-resscn 7934
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-rex 2474  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-pnf 8025
This theorem is referenced by:  renepnf  8036  nn0nepnf  9278  xrltnr  9811  pnfnlt  9819  xnn0lenn0nn0  9897  inftonninf  10474  pcgcd1  12363  pc2dvds  12365
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