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Theorem pnfnre 7998
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 7934 . . . . . 6 ℂ ∈ V
21uniex 4437 . . . . 5 ℂ ∈ V
3 pwuninel2 6282 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 5 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 7993 . . . . 5 +∞ = 𝒫
65eleq1i 2243 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 671 . . 3 ¬ +∞ ∈ ℂ
8 recn 7943 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 662 . 2 ¬ +∞ ∈ ℝ
109nelir 2445 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2148  wnel 2442  Vcvv 2737  𝒫 cpw 3575   cuni 3809  cc 7808  cr 7809  +∞cpnf 7988
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-un 4433  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-nel 2443  df-rex 2461  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810  df-pnf 7993
This theorem is referenced by:  renepnf  8004  nn0nepnf  9246  xrltnr  9778  pnfnlt  9786  xnn0lenn0nn0  9864  inftonninf  10440  pcgcd1  12326  pc2dvds  12328
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