Theorem pnf0xnn0 12511

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 2737	. . 3 ⊢ +∞ = +∞
2	1	olci 867	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12506	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 231	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 848   = wceq 1542   ∈ wcel 2114  +∞cpnf 11170  ℕ₀cn0 12431  ℕ₀^*cxnn0 12504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pow 5303  ax-un 7683  ax-cnex 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-pw 4544  df-sn 4569  df-uni 4852  df-pnf 11175  df-xnn0 12505

This theorem is referenced by:  xnn0xaddcl  13181  pcxnn0cl  16825