Metamath Proof Explorer

Theorem ssriv

Description: Inference based on subclass definition. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypothesis ssriv.1 
|- ( x e. A -> x e. B )
Assertion ssriv 
|- A C_ B

Proof

Step Hyp Ref Expression
1 ssriv.1 
 |-  ( x e. A -> x e. B )
2 df-ss 
 |-  ( A C_ B <-> A. x ( x e. A -> x e. B ) )
3 2 1 mpgbir 
 |-  A C_ B