Theorem pnf0xnn0 12553

Description: Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)

Assertion

Ref	Expression
pnf0xnn0	⊢ +∞ ∈ ℕ0*

Proof of Theorem pnf0xnn0

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ +∞ = +∞
2	1	olci 864	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12548	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 230	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 845   = wceq 1541   ∈ wcel 2106  +∞cpnf 11247  ℕ₀cn0 12474  ℕ₀^*cxnn0 12546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-pow 5363  ax-un 7727  ax-cnex 11168

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-uni 4909  df-pnf 11252  df-xnn0 12547

This theorem is referenced by:  xnn0xaddcl  13216  pcxnn0cl  16795