Theorem pnf0xnn0 11784

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 2771	. . 3 ⊢ +∞ = +∞
2	1	olci 853	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 11779	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 223	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 834   = wceq 1508   ∈ wcel 2051  +∞cpnf 10469  ℕ₀cn0 11705  ℕ₀^*cxnn0 11777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-pow 5115  ax-un 7277  ax-cnex 10389

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-rex 3087  df-v 3410  df-un 3827  df-in 3829  df-ss 3836  df-pw 4418  df-sn 4436  df-uni 4709  df-pnf 10474  df-xnn0 11778

This theorem is referenced by:  xnn0xaddcl  12443