This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elsn2g

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. This variation requires only that B , rather than A , be a set. (Contributed by NM, 28-Oct-2003)

Ref Expression
Assertion elsn2g ⊢ B ∈ V → A ∈ B ↔ A = B

Proof

Step Hyp Ref Expression
1 elsni ⊢ A ∈ B → A = B
2 snidg ⊢ B ∈ V → B ∈ B
3 eleq1 ⊢ A = B → A ∈ B ↔ B ∈ B
4 2 3 syl5ibrcom ⊢ B ∈ V → A = B → A ∈ B
5 1 4 impbid2 ⊢ B ∈ V → A ∈ B ↔ A = B