Theorem pnf0xnn0 12556

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 2731	. . 3 ⊢ +∞ = +∞
2	1	olci 863	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12551	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 230	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 844   = wceq 1540   ∈ wcel 2105  +∞cpnf 11250  ℕ₀cn0 12477  ℕ₀^*cxnn0 12549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pow 5363  ax-un 7729  ax-cnex 11170

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-uni 4909  df-pnf 11255  df-xnn0 12550

This theorem is referenced by:  xnn0xaddcl  13219  pcxnn0cl  16798