Metamath Proof Explorer

Theorem ssinss1

Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026)

Ref Expression
Assertion ssinss1 
|- ( A C_ C -> ( A i^i B ) C_ C )

Proof

Step Hyp Ref Expression
1 ssrin 
 |-  ( A C_ C -> ( A i^i B ) C_ ( C i^i B ) )
2 inss1 
 |-  ( C i^i B ) C_ C
3 1 2 sstrdi 
 |-  ( A C_ C -> ( A i^i B ) C_ C )