Metamath Proof Explorer

Theorem conss2

Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011)

Ref Expression
Assertion conss2 
|- ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) )

Proof

Step Hyp Ref Expression
1 ssv 
 |-  A C_ _V
2 ssv 
 |-  B C_ _V
3 ssconb 
 |-  ( ( A C_ _V /\ B C_ _V ) -> ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) ) )
4 1 2 3 mp2an 
 |-  ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) )