Theorem pnf0xnn0 12461

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 2731	. . 3 ⊢ +∞ = +∞
2	1	olci 866	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12456	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 231	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 847   = wceq 1541   ∈ wcel 2111  +∞cpnf 11143  ℕ₀cn0 12381  ℕ₀^*cxnn0 12454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-pow 5301  ax-un 7668  ax-cnex 11062

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-pw 4549  df-sn 4574  df-uni 4857  df-pnf 11148  df-xnn0 12455

This theorem is referenced by:  xnn0xaddcl  13134  pcxnn0cl  16772