Theorem pnf0xnn0 12603

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 2726	. . 3 ⊢ +∞ = +∞
2	1	olci 864	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12598	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 230	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 845   = wceq 1534   ∈ wcel 2099  +∞cpnf 11295  ℕ₀cn0 12524  ℕ₀^*cxnn0 12596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-pow 5369  ax-un 7746  ax-cnex 11214

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-un 3952  df-ss 3964  df-pw 4609  df-sn 4634  df-uni 4914  df-pnf 11300  df-xnn0 12597

This theorem is referenced by:  xnn0xaddcl  13268  pcxnn0cl  16862