Theorem elnn

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

	Ref	Expression
Assertion	elnn	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω )

Proof

Step	Hyp	Ref	Expression
1		ordom	⊢ Ord ω
2		ordtr	⊢ ( Ord ω → Tr ω )
3		trel	⊢ ( Tr ω → ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω ) )
4	1 2 3	mp2b	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω )