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Metamath Proof Explorer

Theorem bastg

Description: A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006) (Revised by Mario Carneiro, 10-Jan-2015)

Ref Expression
Assertion bastg 
|- ( B e. V -> B C_ ( topGen ` B ) )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( B e. V /\ x e. B ) -> x e. B )
2 vex 
 |-  x e. _V
3 2 pwid 
 |-  x e. ~P x
4 3 a1i 
 |-  ( ( B e. V /\ x e. B ) -> x e. ~P x )
5 1 4 elind 
 |-  ( ( B e. V /\ x e. B ) -> x e. ( B i^i ~P x ) )
6 elssuni 
 |-  ( x e. ( B i^i ~P x ) -> x C_ U. ( B i^i ~P x ) )
7 5 6 syl 
 |-  ( ( B e. V /\ x e. B ) -> x C_ U. ( B i^i ~P x ) )
8 7 ex 
 |-  ( B e. V -> ( x e. B -> x C_ U. ( B i^i ~P x ) ) )
9 eltg 
 |-  ( B e. V -> ( x e. ( topGen ` B ) <-> x C_ U. ( B i^i ~P x ) ) )
10 8 9 sylibrd 
 |-  ( B e. V -> ( x e. B -> x e. ( topGen ` B ) ) )
11 10 ssrdv 
 |-  ( B e. V -> B C_ ( topGen ` B ) )