Theorem nnm0

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

	Ref	Expression
Assertion	nnm0	$$⊢A\in \mathrm{\omega }\to A{\cdot }_{𝑜}\varnothing =\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		nnon	$$⊢A\in \mathrm{\omega }\to A\in \mathrm{On}$$
2		om0	$$⊢A\in \mathrm{On}\to A{\cdot }_{𝑜}\varnothing =\varnothing$$
3	1 2	syl	$$⊢A\in \mathrm{\omega }\to A{\cdot }_{𝑜}\varnothing =\varnothing$$