Theorem mnfnre 8200

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 8134	. . . . 5 ⊢ ℂ ∈ V
2		2pwuninelg 6435	. . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ
4		df-mnf 8195	. . . . . 6 ⊢ -∞ = 𝒫 +∞
5		df-pnf 8194	. . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ
6	5	pweqi 3653	. . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ
7	4, 6	eqtri 2250	. . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ
8	7	eleq1i 2295	. . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ)
9	3, 8	mtbir 675	. . 3 ⊢ ¬ -∞ ∈ ℂ
10		recn 8143	. . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ)
11	9, 10	mto 666	. 2 ⊢ ¬ -∞ ∈ ℝ
12	11	nelir 2498	1 ⊢ -∞ ∉ ℝ

Syntax hints:  ¬ wn 3   ∈ wcel 2200   ∉ wnel 2495  Vcvv 2799  𝒫 cpw 3649  ∪ cuni 3888  ℂcc 8008  ℝcr 8009  +∞cpnf 8189  -∞cmnf 8190

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629  ax-cnex 8101  ax-resscn 8102

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-pnf 8194  df-mnf 8195

This theorem is referenced by:  renemnf  8206  xrltnr  9987  nltmnf  9996