Theorem pnf0xnn0 11247

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 2610	. . 3 ⊢ +∞ = +∞
2	1	olci 405	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 11242	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 220	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977  +∞cpnf 9950  ℕ₀cn0 11169  ℕ₀^*cxnn0 11240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-pow 4769  ax-un 6847  ax-cnex 9871

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-pnf 9955  df-xr 9957  df-xnn0 11241

This theorem is referenced by:  xnn0xaddcl  11940