Theorem mnfnre 8114

Description: Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
mnfnre	⊢ -∞ ∉ ℝ

Proof of Theorem mnfnre

Step	Hyp	Ref	Expression
1		cnex 8048	. . . . 5 ⊢ ℂ ∈ V
2		2pwuninelg 6368	. . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ
4		df-mnf 8109	. . . . . 6 ⊢ -∞ = 𝒫 +∞
5		df-pnf 8108	. . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	pweqi 3619	. . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ
7	4, 6	eqtri 2225	. . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ
8	7	eleq1i 2270	. . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
9	3, 8	mtbir 672	. . 3 ⊢ ¬ -∞ ∈ ℂ
10		recn 8057	. . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ)
11	9, 10	mto 663	. 2 ⊢ ¬ -∞ ∈ ℝ
12	11	nelir 2473	1 ⊢ -∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2175   ∉ wnel 2470  Vcvv 2771  𝒫 cpw 3615  ∪ cuni 3849  ℂcc 7922  ℝcr 7923  +∞cpnf 8103  -∞cmnf 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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-setind 4584  ax-cnex 8015  ax-resscn 8016

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-nel 2471  df-ral 2488  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-pnf 8108  df-mnf 8109

This theorem is referenced by:  renemnf  8120  xrltnr  9900  nltmnf  9909