Theorem mnfnre 7068

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 7005	. . . . 5 ⊢ ℂ ∈ V
2		2pwuninelg 5898	. . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
3	1, 2	ax-mp 7	. . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ
4		df-mnf 7063	. . . . . 6 ⊢ -∞ = 𝒫 +∞
5		df-pnf 7062	. . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	pweqi 3363	. . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ
7	4, 6	eqtri 2060	. . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ
8	7	eleq1i 2103	. . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
9	3, 8	mtbir 596	. . 3 ⊢ ¬ -∞ ∈ ℂ
10		recn 7014	. . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ)
11	9, 10	mto 588	. 2 ⊢ ¬ -∞ ∈ ℝ
12	11	nelir 2300	1 ⊢ -∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 1393   ∉ wnel 2205  Vcvv 2557  𝒫 cpw 3359  ∪ cuni 3580  ℂcc 6887  ℝcr 6888  +∞cpnf 7057  -∞cmnf 7058

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262  ax-cnex 6975  ax-resscn 6976

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7062  df-mnf 7063

This theorem is referenced by:  renemnf  7074  xrltnr  8701  nltmnf  8709