Theorem pclssidN

Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclss.a	|- A = ( Atoms ` K )
	pclss.c	|- U = ( PCl ` K )
Assertion	pclssidN	|- ( ( K e. V /\ X C_ A ) -> X C_ ( U ` X ) )

Proof

Step	Hyp	Ref	Expression
1		pclss.a	|- A = ( Atoms ` K )
2		pclss.c	|- U = ( PCl ` K )
3		ssintub	|- X C_ |^| { y e. ( PSubSp ` K ) | X C_ y }
4		eqid	|- ( PSubSp ` K ) = ( PSubSp ` K )
5	1 4 2	pclvalN	|- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. ( PSubSp ` K ) | X C_ y } )
6	3 5	sseqtrrid	|- ( ( K e. V /\ X C_ A ) -> X C_ ( U ` X ) )