Theorem pnfnre 8188

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 8123	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4528	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6428	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8183	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2295	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 675	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8132	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 666	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2498	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2200   ∉ wnel 2495  Vcvv 2799  𝒫 cpw 3649  ∪ cuni 3888  ℂcc 7997  ℝcr 7998  +∞cpnf 8178

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-un 4524  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-rex 2514  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-uni 3889  df-pnf 8183

This theorem is referenced by:  renepnf  8194  nn0nepnf  9440  xrltnr  9975  pnfnlt  9983  xnn0lenn0nn0  10061  inftonninf  10664  pcgcd1  12851  pc2dvds  12853