Metamath Proof Explorer

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