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Theorem pnfnre 7934
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 7871 . . . . . 6 ℂ ∈ V
21uniex 4412 . . . . 5 ℂ ∈ V
3 pwuninel2 6244 . . . . 5 ( ℂ ∈ V → ¬ 𝒫 ℂ ∈ ℂ)
42, 3ax-mp 5 . . . 4 ¬ 𝒫 ℂ ∈ ℂ
5 df-pnf 7929 . . . . 5 +∞ = 𝒫
65eleq1i 2230 . . . 4 (+∞ ∈ ℂ ↔ 𝒫 ℂ ∈ ℂ)
74, 6mtbir 661 . . 3 ¬ +∞ ∈ ℂ
8 recn 7880 . . 3 (+∞ ∈ ℝ → +∞ ∈ ℂ)
97, 8mto 652 . 2 ¬ +∞ ∈ ℝ
109nelir 2432 1 +∞ ∉ ℝ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2135  wnel 2429  Vcvv 2724  𝒫 cpw 3556   cuni 3786  cc 7745  cr 7746  +∞cpnf 7924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-un 4408  ax-cnex 7838  ax-resscn 7839
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-nel 2430  df-rex 2448  df-rab 2451  df-v 2726  df-in 3120  df-ss 3127  df-pw 3558  df-uni 3787  df-pnf 7929
This theorem is referenced by:  renepnf  7940  nn0nepnf  9179  xrltnr  9709  pnfnlt  9717  xnn0lenn0nn0  9795  inftonninf  10370  pcgcd1  12253  pc2dvds  12255
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