Theorem pnf0xnn0 12632

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 2740	. . 3 ⊢ +∞ = +∞
2	1	olci 865	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12627	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 231	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 846   = wceq 1537   ∈ wcel 2108  +∞cpnf 11321  ℕ₀cn0 12553  ℕ₀^*cxnn0 12625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pow 5383  ax-un 7770  ax-cnex 11240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-pw 4624  df-sn 4649  df-uni 4932  df-pnf 11326  df-xnn0 12626

This theorem is referenced by:  xnn0xaddcl  13297  pcxnn0cl  16907