Theorem elsuc2

Description: Membership in a successor. (Contributed by NM, 15-Sep-2003)

	Ref	Expression
Hypothesis	elsuc.1	⊢ 𝐴 ∈ V
Assertion	elsuc2	⊢ ( 𝐵 ∈ suc 𝐴 ↔ ( 𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elsuc.1	⊢ 𝐴 ∈ V
2		elsuc2g	⊢ ( 𝐴 ∈ V → ( 𝐵 ∈ suc 𝐴 ↔ ( 𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( 𝐵 ∈ suc 𝐴 ↔ ( 𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴 ) )