Theorem mnfnre 8332

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 8267	. . . . 5 ⊢ ℂ ∈ V
2		2pwuninelg 6527	. . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ
4		df-mnf 8327	. . . . . 6 ⊢ -∞ = 𝒫 +∞
5		df-pnf 8326	. . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	pweqi 3678	. . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ
7	4, 6	eqtri 2255	. . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ
8	7	eleq1i 2300	. . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
9	3, 8	mtbir 678	. . 3 ⊢ ¬ -∞ ∈ ℂ
10		recn 8276	. . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ)
11	9, 10	mto 668	. 2 ⊢ ¬ -∞ ∈ ℝ
12	11	nelir 2512	1 ⊢ -∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2205   ∉ wnel 2509  Vcvv 2815  𝒫 cpw 3674  ∪ cuni 3919  ℂcc 8141  ℝcr 8142  +∞cpnf 8321  -∞cmnf 8322

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-setind 4664  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-nel 2510  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-pnf 8326  df-mnf 8327

This theorem is referenced by:  renemnf  8338  xrltnr  10131  nltmnf  10140