Theorem pnfnre 8315

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 8251	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4558	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6513	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8310	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2298	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 678	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8260	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 668	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2510	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2203   ∉ wnel 2507  Vcvv 2813  𝒫 cpw 3669  ∪ cuni 3914  ℂcc 8125  ℝcr 8126  +∞cpnf 8305

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-nel 2508  df-rex 2526  df-rab 2529  df-v 2815  df-in 3217  df-ss 3224  df-pw 3671  df-uni 3915  df-pnf 8310

This theorem is referenced by:  renepnf  8321  nn0nepnf  9571  xrltnr  10112  pnfnlt  10120  xnn0lenn0nn0  10198  inftonninf  10804  pcgcd1  13026  pc2dvds  13028