Theorem pnfnre 8001

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 7937	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4439	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6285	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 7996	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2243	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 671	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 7946	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 662	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2445	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2148   ∉ wnel 2442  Vcvv 2739  𝒫 cpw 3577  ∪ cuni 3811  ℂcc 7811  ℝcr 7812  +∞cpnf 7991

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 4123  ax-un 4435  ax-cnex 7904  ax-resscn 7905

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 2741  df-in 3137  df-ss 3144  df-pw 3579  df-uni 3812  df-pnf 7996

This theorem is referenced by:  renepnf  8007  nn0nepnf  9249  xrltnr  9781  pnfnlt  9789  xnn0lenn0nn0  9867  inftonninf  10443  pcgcd1  12329  pc2dvds  12331