Theorem pnfnre 8331

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 8267	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4563	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6526	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8326	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2300	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 678	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8276	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 668	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2512	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2205   ∉ wnel 2509  Vcvv 2815  𝒫 cpw 3674  ∪ cuni 3919  ℂcc 8141  ℝcr 8142  +∞cpnf 8321

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-nel 2510  df-rex 2528  df-rab 2531  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676  df-uni 3920  df-pnf 8326

This theorem is referenced by:  renepnf  8337  nn0nepnf  9588  xrltnr  10131  pnfnlt  10139  xnn0lenn0nn0  10217  inftonninf  10828  pcgcd1  13051  pc2dvds  13053