Theorem elelsuc

Description: Membership in a successor. (Contributed by NM, 20-Jun-1998)

	Ref	Expression
Assertion	elelsuc	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) )
2		elsucg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ suc 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
3	1 2	mpbird	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵 )