Theorem pnf0xnn0 11370

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 2622	. . 3 ⊢ +∞ = +∞
2	1	olci 406	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 11365	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 221	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 383   = wceq 1483   ∈ wcel 1990  +∞cpnf 10071  ℕ₀cn0 11292  ℕ₀^*cxnn0 11363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-pnf 10076  df-xr 10078  df-xnn0 11364

This theorem is referenced by:  xnn0xaddcl  12066