Metamath Proof Explorer


Theorem sels

Description: If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002) Generalize from the proof of elALT . (Revised by BJ, 3-Apr-2019) Avoid ax-sep , ax-nul , ax-pow . (Revised by BTernaryTau, 15-Jan-2025)

Ref Expression
Assertion sels
|- ( A e. V -> E. x A e. x )

Proof

Step Hyp Ref Expression
1 eleq1
 |-  ( y = A -> ( y e. x <-> A e. x ) )
2 1 exbidv
 |-  ( y = A -> ( E. x y e. x <-> E. x A e. x ) )
3 el
 |-  E. x y e. x
4 2 3 vtoclg
 |-  ( A e. V -> E. x A e. x )