Theorem elxnn0

Description: An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020)

	Ref	Expression
Assertion	elxnn0	⊢ ( 𝐴 ∈ ℕ0* ↔ ( 𝐴 ∈ ℕ0 ∨ 𝐴 = +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		df-xnn0	⊢ ℕ0* = ( ℕ0 ∪ { +∞ } )
2	1	eleq2i	⊢ ( 𝐴 ∈ ℕ0* ↔ 𝐴 ∈ ( ℕ0 ∪ { +∞ } ) )
3		elun	⊢ ( 𝐴 ∈ ( ℕ0 ∪ { +∞ } ) ↔ ( 𝐴 ∈ ℕ0 ∨ 𝐴 ∈ { +∞ } ) )
4		pnfex	⊢ +∞ ∈ V
5	4	elsn2	⊢ ( 𝐴 ∈ { +∞ } ↔ 𝐴 = +∞ )
6	5	orbi2i	⊢ ( ( 𝐴 ∈ ℕ0 ∨ 𝐴 ∈ { +∞ } ) ↔ ( 𝐴 ∈ ℕ0 ∨ 𝐴 = +∞ ) )
7	2 3 6	3bitri	⊢ ( 𝐴 ∈ ℕ0* ↔ ( 𝐴 ∈ ℕ0 ∨ 𝐴 = +∞ ) )