Theorem mnfnre 8316

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 8251	. . . . 5 ⊢ ℂ ∈ V
2		2pwuninelg 6514	. . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ
4		df-mnf 8311	. . . . . 6 ⊢ -∞ = 𝒫 +∞
5		df-pnf 8310	. . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	pweqi 3673	. . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ
7	4, 6	eqtri 2253	. . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ
8	7	eleq1i 2298	. . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
9	3, 8	mtbir 678	. . 3 ⊢ ¬ -∞ ∈ ℂ
10		recn 8260	. . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ)
11	9, 10	mto 668	. 2 ⊢ ¬ -∞ ∈ ℝ
12	11	nelir 2510	1 ⊢ -∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2203   ∉ wnel 2507  Vcvv 2813  𝒫 cpw 3669  ∪ cuni 3914  ℂcc 8125  ℝcr 8126  +∞cpnf 8305  -∞cmnf 8306

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 2214  ax-setind 4659  ax-cnex 8218  ax-resscn 8219

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-nel 2508  df-ral 2525  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-pnf 8310  df-mnf 8311

This theorem is referenced by:  renemnf  8322  xrltnr  10112  nltmnf  10121