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Metamath Proof Explorer

Theorem snssb

Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025)

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

Proof

Step Hyp Ref Expression
1 df-ss 
 |-  ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2 velsn 
 |-  ( x e. { A } <-> x = A )
3 2 imbi1i 
 |-  ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
4 3 albii 
 |-  ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
5 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
6 5 pm5.74i 
 |-  ( ( x = A -> x e. B ) <-> ( x = A -> A e. B ) )
7 6 albii 
 |-  ( A. x ( x = A -> x e. B ) <-> A. x ( x = A -> A e. B ) )
8 19.23v 
 |-  ( A. x ( x = A -> A e. B ) <-> ( E. x x = A -> A e. B ) )
9 isset 
 |-  ( A e. _V <-> E. x x = A )
10 9 bicomi 
 |-  ( E. x x = A <-> A e. _V )
11 10 imbi1i 
 |-  ( ( E. x x = A -> A e. B ) <-> ( A e. _V -> A e. B ) )
12 7 8 11 3bitri 
 |-  ( A. x ( x = A -> x e. B ) <-> ( A e. _V -> A e. B ) )
13 1 4 12 3bitri 
 |-  ( { A } C_ B <-> ( A e. _V -> A e. B ) )