Theorem sucssel

Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995)

	Ref	Expression
Assertion	sucssel	⊢ ( 𝐴 ∈ 𝑉 → ( suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sucidg	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴 )
2		ssel	⊢ ( suc 𝐴 ⊆ 𝐵 → ( 𝐴 ∈ suc 𝐴 → 𝐴 ∈ 𝐵 ) )
3	1 2	syl5com	⊢ ( 𝐴 ∈ 𝑉 → ( suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵 ) )