Theorem addclpi

Description: Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	addclpi	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) ∈ N )

Proof

Step	Hyp	Ref	Expression
1		addpiord	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )
2		pinn	⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
3		pinn	⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
4		nnacl	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o 𝐵 ) ∈ ω )
5	3 4	sylan2	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ ω )
6		elni2	⊢ ( 𝐵 ∈ N ↔ ( 𝐵 ∈ ω ∧ ∅ ∈ 𝐵 ) )
7		nnaordi	⊢ ( ( 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) → ( ∅ ∈ 𝐵 → ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
8		ne0i	⊢ ( ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
9	7 8	syl6	⊢ ( ( 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) → ( ∅ ∈ 𝐵 → ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
10	9	expcom	⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( ∅ ∈ 𝐵 → ( 𝐴 +o 𝐵 ) ≠ ∅ ) ) )
11	10	imp32	⊢ ( ( 𝐴 ∈ ω ∧ ( 𝐵 ∈ ω ∧ ∅ ∈ 𝐵 ) ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
12	6 11	sylan2b	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
13		elni	⊢ ( ( 𝐴 +o 𝐵 ) ∈ N ↔ ( ( 𝐴 +o 𝐵 ) ∈ ω ∧ ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
14	5 12 13	sylanbrc	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ N )
15	2 14	sylan	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ N )
16	1 15	eqeltrd	⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) ∈ N )