Theorem pnfnre 8063

Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
pnfnre	⊢ +∞ ∉ ℝ

Proof of Theorem pnfnre

Step	Hyp	Ref	Expression
1		cnex 7998	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4469	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6337	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8058	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2259	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 672	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8007	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 663	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2462	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2164   ∉ wnel 2459  Vcvv 2760  𝒫 cpw 3602  ∪ cuni 3836  ℂcc 7872  ℝcr 7873  +∞cpnf 8053

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-un 4465  ax-cnex 7965  ax-resscn 7966

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-nel 2460  df-rex 2478  df-rab 2481  df-v 2762  df-in 3160  df-ss 3167  df-pw 3604  df-uni 3837  df-pnf 8058

This theorem is referenced by:  renepnf  8069  nn0nepnf  9314  xrltnr  9848  pnfnlt  9856  xnn0lenn0nn0  9934  inftonninf  10516  pcgcd1  12469  pc2dvds  12471