Theorem pnfnre 8114

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 8049	. . . . . 6 ⊢ ℂ ∈ V
2	1	uniex 4484	. . . . 5 ⊢ ∪ ℂ ∈ V
3		pwuninel2 6368	. . . . 5 ⊢ (∪ ℂ ∈ V → ¬ 𝒫 ∪ ℂ ∈ ℂ)
4	2, 3	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
5		df-pnf 8109	. . . . 5 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	eleq1i 2271	. . . 4 ⊢ (+∞ ∈ ℂ ↔ 𝒫 ∪ ℂ ∈ ℂ)
7	4, 6	mtbir 673	. . 3 ⊢ ¬ +∞ ∈ ℂ
8		recn 8058	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
9	7, 8	mto 664	. 2 ⊢ ¬ +∞ ∈ ℝ
10	9	nelir 2474	1 ⊢ +∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2176   ∉ wnel 2471  Vcvv 2772  𝒫 cpw 3616  ∪ cuni 3850  ℂcc 7923  ℝcr 7924  +∞cpnf 8104

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-un 4480  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-nel 2472  df-rex 2490  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-uni 3851  df-pnf 8109

This theorem is referenced by:  renepnf  8120  nn0nepnf  9366  xrltnr  9901  pnfnlt  9909  xnn0lenn0nn0  9987  inftonninf  10587  pcgcd1  12651  pc2dvds  12653