Theorem pnf0xnn0 12464

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 2729	. . 3 ⊢ +∞ = +∞
2	1	olci 866	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12459	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 231	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 847   = wceq 1540   ∈ wcel 2109  +∞cpnf 11146  ℕ₀cn0 12384  ℕ₀^*cxnn0 12457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pow 5304  ax-un 7671  ax-cnex 11065

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-un 3908  df-ss 3920  df-pw 4553  df-sn 4578  df-uni 4859  df-pnf 11151  df-xnn0 12458

This theorem is referenced by:  xnn0xaddcl  13137  pcxnn0cl  16772