Theorem naddlid

Description: Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025)

	Ref	Expression
Assertion	naddlid	⊢ ( 𝐴 ∈ On → ( ∅ +no 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		0elon	⊢ ∅ ∈ On
2		naddcom	⊢ ( ( 𝐴 ∈ On ∧ ∅ ∈ On ) → ( 𝐴 +no ∅ ) = ( ∅ +no 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ On → ( 𝐴 +no ∅ ) = ( ∅ +no 𝐴 ) )
4		naddrid	⊢ ( 𝐴 ∈ On → ( 𝐴 +no ∅ ) = 𝐴 )
5	3 4	eqtr3d	⊢ ( 𝐴 ∈ On → ( ∅ +no 𝐴 ) = 𝐴 )