Theorem pnfnre 8357

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 8293	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4578	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6543	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8352	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2304	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 682	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8302	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 672	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2518	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2209   ∉ wnel 2515  Vcvv 2821  𝒫 cpw 3685  ∪ cuni 3930  ℂcc 8167  ℝcr 8168  +∞cpnf 8347

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-un 4573  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-nel 2516  df-rex 2534  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931  df-pnf 8352

This theorem is referenced by:  renepnf  8363  nn0nepnf  9617  xrltnr  10160  pnfnlt  10168  xnn0lenn0nn0  10246  inftonninf  10857  pcgcd1  13085  pc2dvds  13087