Theorem pnfnre 11374

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 11369	. . . 4 ⊢ +∞ = 𝒫 ∪ ℂ
2		pwuninel 8317	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
3	1, 2	eqneltri 2883	. . 3 ⊢ ¬ +∞ ∈ ℂ
4		recn 11314	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
5	3, 4	mto 200	. 2 ⊢ ¬ +∞ ∈ ℝ
6	5	nelir 3068	1 ⊢ +∞ ∉ ℝ

This proof depends on syntax axioms:   ∈ wcel 2147   ∉ wnel 3065  𝒫 cpw 4560  ∪ cuni 4870  ℂcc 11222  ℝcr 11223  +∞cpnf 11364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-resscn 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nel 3066  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-uni 4871  df-pnf 11369

This theorem is used by:  pnfnre2  11375  renepnf  11381  ltxrlt  11404  nn0nepnf  12711  xrltnr  13272  pnfnlt  13281  xnn0lenn0nn0  13399  hashclb  14526  hasheq0  14531  pcgcd1  17081  pc2dvds  17083  ramtcl2  17215  odhash3  19817  xrsdsreclblem  21746  pnfnei  23565  iccpnfcnv  25292  i1f0rn  26030