Metamath Proof Explorer


Theorem snssgOLD

Description: Obsolete version of snssgOLD as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion snssgOLD
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Proof

Step Hyp Ref Expression
1 velsn
 |-  ( x e. { A } <-> x = A )
2 1 imbi1i
 |-  ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
3 2 albii
 |-  ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4 3 a1i
 |-  ( A e. V -> ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) ) )
5 dfss2
 |-  ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
6 5 a1i
 |-  ( A e. V -> ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) ) )
7 clel2g
 |-  ( A e. V -> ( A e. B <-> A. x ( x = A -> x e. B ) ) )
8 4 6 7 3bitr4rd
 |-  ( A e. V -> ( A e. B <-> { A } C_ B ) )