Theorem elnn

Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998)

	Ref	Expression
Assertion	elnn	$$⊢\left(A\in B\wedge B\in \mathrm{\omega }\right)\to A\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		trom	$$⊢\mathrm{Tr}\mathrm{\omega }$$
2		trel	$$⊢\mathrm{Tr}\mathrm{\omega }\to \left(\left(A\in B\wedge B\in \mathrm{\omega }\right)\to A\in \mathrm{\omega }\right)$$
3	1 2	ax-mp	$$⊢\left(A\in B\wedge B\in \mathrm{\omega }\right)\to A\in \mathrm{\omega }$$