Theorem pnf0xnn0 12026

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 2758	. . 3 ⊢ +∞ = +∞
2	1	olci 863	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12021	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 844   = wceq 1538   ∈ wcel 2111  +∞cpnf 10723  ℕ₀cn0 11947  ℕ₀^*cxnn0 12019

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-pow 5238  ax-un 7465  ax-cnex 10644

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-pw 4499  df-sn 4526  df-uni 4802  df-pnf 10728  df-xnn0 12020

This theorem is referenced by:  xnn0xaddcl  12682