Theorem pnfnre 11255

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 11250	. . . 4 ⊢ +∞ = 𝒫 ∪ ℂ
2		pwuninel 8260	. . . 4 ⊢ ¬ 𝒫 ∪ ℂ ∈ ℂ
3	1, 2	eqneltri 2853	. . 3 ⊢ ¬ +∞ ∈ ℂ
4		recn 11200	. . 3 ⊢ (+∞ ∈ ℝ → +∞ ∈ ℂ)
5	3, 4	mto 196	. 2 ⊢ ¬ +∞ ∈ ℝ
6	5	nelir 3050	1 ⊢ +∞ ∉ ℝ

Syntax hints:   ∈ wcel 2107   ∉ wnel 3047  𝒫 cpw 4603  ∪ cuni 4909  ℂcc 11108  ℝcr 11109  +∞cpnf 11245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-pr 5428  ax-un 7725  ax-resscn 11167

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3048  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-pnf 11250

This theorem is referenced by:  pnfnre2  11256  renepnf  11262  ltxrlt  11284  nn0nepnf  12552  xrltnr  13099  pnfnlt  13108  xnn0lenn0nn0  13224  hashclb  14318  hasheq0  14323  pcgcd1  16810  pc2dvds  16812  ramtcl2  16944  odhash3  19444  xrsdsreclblem  20991  pnfnei  22724  iccpnfcnv  24460  i1f0rn  25199