Theorem pnfnre 8220

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 8155	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4534	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6447	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8215	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2297	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 677	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8164	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 668	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2500	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2202   ∉ wnel 2497  Vcvv 2802  𝒫 cpw 3652  ∪ cuni 3893  ℂcc 8029  ℝcr 8030  +∞cpnf 8210

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-un 4530  ax-cnex 8122  ax-resscn 8123

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-nel 2498  df-rex 2516  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-uni 3894  df-pnf 8215

This theorem is referenced by:  renepnf  8226  nn0nepnf  9472  xrltnr  10013  pnfnlt  10021  xnn0lenn0nn0  10099  inftonninf  10703  pcgcd1  12900  pc2dvds  12902