Theorem pnf0xnn0 12583

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 2769	. . 3 ⊢ +∞ = +∞
2	1	olci 879	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12578	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 860   = wceq 1567   ∈ wcel 2149  +∞cpnf 11239  ℕ₀cn0 12503  ℕ₀^*cxnn0 12576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-un 7733  ax-cnex 11155

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-pw 4569  df-sn 4595  df-uni 4877  df-pnf 11244  df-xnn0 12577

This theorem is referenced by:  xnn0xaddcl  13260  pcxnn0cl  16919