Theorem polssatN

Description: The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	polssat.a	|- A = ( Atoms ` K )
	polssat.p	|- ._|_ = ( _|_P ` K )
Assertion	polssatN	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )

Proof

Step	Hyp	Ref	Expression
1		polssat.a	|- A = ( Atoms ` K )
2		polssat.p	|- ._|_ = ( _|_P ` K )
3		eqid	|- ( PSubSp ` K ) = ( PSubSp ` K )
4	1 3 2	polsubN	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. ( PSubSp ` K ) )
5	1 3	psubssat	|- ( ( K e. HL /\ ( ._|_ ` X ) e. ( PSubSp ` K ) ) -> ( ._|_ ` X ) C_ A )
6	4 5	syldan	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )