Theorem pnf0xnn0 12561

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 2762	. . 3 ⊢ +∞ = +∞
2	1	olci 877	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12556	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 233	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 858   = wceq 1560   ∈ wcel 2142  +∞cpnf 11213  ℕ₀cn0 12481  ℕ₀^*cxnn0 12554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-un 7718  ax-cnex 11129

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4557  df-sn 4583  df-uni 4866  df-pnf 11218  df-xnn0 12555

This theorem is referenced by:  xnn0xaddcl  13238  pcxnn0cl  16896