Theorem naddcld

Description: Closure law for natural addition. Deduction version. (Contributed by Scott Fenton, 10-Jun-2025)

	Ref	Expression
Hypotheses	naddcld.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	naddcld.2	⊢ ( 𝜑 → 𝐵 ∈ On )
Assertion	naddcld	⊢ ( 𝜑 → ( 𝐴 +no 𝐵 ) ∈ On )

Proof

Step	Hyp	Ref	Expression
1		naddcld.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		naddcld.2	⊢ ( 𝜑 → 𝐵 ∈ On )
3		naddcl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) ∈ On )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +no 𝐵 ) ∈ On )