Theorem pnf0xnn0 12579

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 2763	. . 3 ⊢ +∞ = +∞
2	1	olci 879	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12574	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 860   = wceq 1570   ∈ wcel 2143  +∞cpnf 11235  ℕ₀cn0 12499  ℕ₀^*cxnn0 12572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-un 7732  ax-cnex 11151

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-ss 3922  df-pw 4564  df-sn 4590  df-uni 4873  df-pnf 11240  df-xnn0 12573

This theorem is referenced by:  xnn0xaddcl  13256  pcxnn0cl  16915