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Metamath Proof Explorer

Theorem elsn

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. (Contributed by NM, 13-Sep-1995)

Ref Expression
Hypothesis elsn.1 ⊢ A ∈ V
Assertion elsn ⊢ A ∈ B ↔ A = B

Proof

Step Hyp Ref Expression
1 elsn.1 ⊢ A ∈ V
2 elsng ⊢ A ∈ V → A ∈ B ↔ A = B
3 1 2 ax-mp ⊢ A ∈ B ↔ A = B