Theorem pnfnre 11296

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

Assertion

Ref	Expression
pnfnre	⊢ +∞ ∉ ℝ

Proof of Theorem pnfnre

Step	Hyp	Ref	Expression
1		df-pnf 11291	. . . 4 ⊢ +∞ = 𝒫 ∪ ℂ
2		pwuninel 8282	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
3	1, 2	eqneltri 2845	. . 3 ⊢ ¬ +∞ ∈ ℂ
4		recn 11239	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
5	3, 4	mto 196	. 2 ⊢ ¬ +∞ ∈ ℝ
6	5	nelir 3039	1 ⊢ +∞ ∉ ℝ

Syntax hints:   ∈ wcel 2099   ∉ wnel 3036  𝒫 cpw 4597  ∪ cuni 4905  ℂcc 11147  ℝcr 11148  +∞cpnf 11286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-pr 5425  ax-un 7738  ax-resscn 11206

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nel 3037  df-rab 3420  df-v 3464  df-un 3951  df-in 3953  df-ss 3963  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4906  df-pnf 11291

This theorem is referenced by:  pnfnre2  11297  renepnf  11303  ltxrlt  11325  nn0nepnf  12598  xrltnr  13147  pnfnlt  13156  xnn0lenn0nn0  13272  hashclb  14370  hasheq0  14375  pcgcd1  16874  pc2dvds  16876  ramtcl2  17008  odhash3  19570  xrsdsreclblem  21405  pnfnei  23212  iccpnfcnv  24957  i1f0rn  25699