Theorem pnf0xnn0 12693

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 2760	. . 3 ⊢ +∞ = +∞
2	1	olci 880	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12688	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1569   ∈ wcel 2144  +∞cpnf 11347  ℕ₀cn0 12613  ℕ₀^*cxnn0 12686

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5327  ax-un 7755  ax-cnex 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-pw 4558  df-sn 4585  df-uni 4867  df-pnf 11352  df-xnn0 12687

This theorem is used by:  xnn0xaddcl  13372  pcxnn0cl  17047