Theorem pnf0xnn0 12508

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 2739	. . 3 ⊢ +∞ = +∞
2	1	olci 872	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12503	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 232	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 853   = wceq 1547   ∈ wcel 2119  +∞cpnf 11167  ℕ₀cn0 12428  ℕ₀^*cxnn0 12501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-un 7678  ax-cnex 11085

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-ss 3900  df-pw 4531  df-sn 4556  df-uni 4839  df-pnf 11172  df-xnn0 12502

This theorem is referenced by:  xnn0xaddcl  13178  pcxnn0cl  16822