Metamath Proof Explorer

Theorem euen1b

Description: Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015)

Ref Expression
Assertion euen1b 
|- ( A ~~ 1o <-> E! x x e. A )

Proof

Step Hyp Ref Expression
1 euen1 
 |-  ( E! x x e. A <-> { x | x e. A } ~~ 1o )
2 abid2 
 |-  { x | x e. A } = A
3 2 breq1i 
 |-  ( { x | x e. A } ~~ 1o <-> A ~~ 1o )
4 1 3 bitr2i 
 |-  ( A ~~ 1o <-> E! x x e. A )