Theorem somin2

Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015)

	Ref	Expression
Assertion	somin2	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) ( R u. _I ) B )

Proof

Step	Hyp	Ref	Expression
1		somincom	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) = if ( B R A , B , A ) )
2		somin1	|- ( ( R Or X /\ ( B e. X /\ A e. X ) ) -> if ( B R A , B , A ) ( R u. _I ) B )
3	2	ancom2s	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( B R A , B , A ) ( R u. _I ) B )
4	1 3	eqbrtrd	|- ( ( R Or X /\ ( A e. X /\ B e. X ) ) -> if ( A R B , A , B ) ( R u. _I ) B )