Theorem pnfnre 11253

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 11248	. . . 4 ⊢ +∞ = 𝒫 ∪ ℂ
2		pwuninel 8274	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
3	1, 2	eqneltri 2889	. . 3 ⊢ ¬ +∞ ∈ ℂ
4		recn 11193	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
5	3, 4	mto 200	. 2 ⊢ ¬ +∞ ∈ ℝ
6	5	nelir 3074	1 ⊢ +∞ ∉ ℝ

Syntax hints:   ∈ wcel 2150   ∉ wnel 3071  𝒫 cpw 4567  ∪ cuni 4877  ℂcc 11101  ℝcr 11102  +∞cpnf 11243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-resscn 11160

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nel 3072  df-rab 3424  df-v 3464  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4878  df-pnf 11248

This theorem is referenced by:  pnfnre2  11254  renepnf  11260  ltxrlt  11283  nn0nepnf  12588  xrltnr  13147  pnfnlt  13156  xnn0lenn0nn0  13274  hashclb  14397  hasheq0  14402  pcgcd1  16940  pc2dvds  16942  ramtcl2  17074  odhash3  19649  xrsdsreclblem  21546  pnfnei  23360  iccpnfcnv  25086  i1f0rn  25824