Theorem psubatN

Description: A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	atpsub.a	|- A = ( Atoms ` K )
	atpsub.s	|- S = ( PSubSp ` K )
Assertion	psubatN	|- ( ( K e. B /\ X e. S /\ Y e. X ) -> Y e. A )

Proof

Step	Hyp	Ref	Expression
1		atpsub.a	|- A = ( Atoms ` K )
2		atpsub.s	|- S = ( PSubSp ` K )
3	1 2	psubssat	|- ( ( K e. B /\ X e. S ) -> X C_ A )
4	3	sseld	|- ( ( K e. B /\ X e. S ) -> ( Y e. X -> Y e. A ) )
5	4	3impia	|- ( ( K e. B /\ X e. S /\ Y e. X ) -> Y e. A )