Theorem nnm0

Description: Multiplication with zero. Theorem 4J(A1) of Enderton p. 80. (Contributed by NM, 20-Sep-1995)

	Ref	Expression
Assertion	nnm0	⊢ ( 𝐴 ∈ ω → ( 𝐴 ·o ∅ ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		nnon	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2		om0	⊢ ( 𝐴 ∈ On → ( 𝐴 ·o ∅ ) = ∅ )
3	1 2	syl	⊢ ( 𝐴 ∈ ω → ( 𝐴 ·o ∅ ) = ∅ )