Theorem elsuc

Description: Membership in a successor. Exercise 5 of TakeutiZaring p. 17. (Contributed by NM, 15-Sep-2003)

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

Proof

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