Theorem pnf0xnn0 12242

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 862	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12237	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 230	1 ⊢ +∞ ∈ ℕ0*

Syntax hints:   ∨ wo 843   = wceq 1539   ∈ wcel 2108  +∞cpnf 10937  ℕ₀cn0 12163  ℕ₀^*cxnn0 12235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-pow 5283  ax-un 7566  ax-cnex 10858

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-pw 4532  df-sn 4559  df-uni 4837  df-pnf 10942  df-xnn0 12236

This theorem is referenced by:  xnn0xaddcl  12898  pcxnn0cl  16489