Theorem pnf0xnn0 12312

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 2738	. . 3 ⊢ +∞ = +∞
2	1	olci 863	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12307	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 230	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 844   = wceq 1539   ∈ wcel 2106  +∞cpnf 11006  ℕ₀cn0 12233  ℕ₀^*cxnn0 12305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-pow 5288  ax-un 7588  ax-cnex 10927

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-pw 4535  df-sn 4562  df-uni 4840  df-pnf 11011  df-xnn0 12306

This theorem is referenced by:  xnn0xaddcl  12969  pcxnn0cl  16561