Theorem pnf0xnn0 12686

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 2761	. . 3 ⊢ +∞ = +∞
2	1	olci 880	. 2 ⊢ (+∞ ∈ ℕ0 ∨ +∞ = +∞)
3		elxnn0 12681	. 2 ⊢ (+∞ ∈ ℕ0* ↔ (+∞ ∈ ℕ0 ∨ +∞ = +∞))
4	2, 3	mpbir 234	1 ⊢ +∞ ∈ ℕ0*

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ∈ wcel 2145  +∞cpnf 11340  ℕ₀cn0 12606  ℕ₀^*cxnn0 12679

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-un 7751  ax-cnex 11256

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-pw 4559  df-sn 4585  df-uni 4868  df-pnf 11345  df-xnn0 12680

This theorem is used by:  xnn0xaddcl  13365  pcxnn0cl  17038