Theorem clatp1cl

Description: The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018)

	Ref	Expression
Hypotheses	clatp1cl.b	|- B = ( Base ` W )
	clatp1cl.1	|- .1. = ( 1. ` W )
Assertion	clatp1cl	|- ( W e. CLat -> .1. e. B )

Proof

Step	Hyp	Ref	Expression
1		clatp1cl.b	|- B = ( Base ` W )
2		clatp1cl.1	|- .1. = ( 1. ` W )
3		eqid	|- ( lub ` W ) = ( lub ` W )
4	1 3 2	p1val	|- ( W e. CLat -> .1. = ( ( lub ` W ) ` B ) )
5		ssid	|- B C_ B
6	1 3	clatlubcl	|- ( ( W e. CLat /\ B C_ B ) -> ( ( lub ` W ) ` B ) e. B )
7	5 6	mpan2	|- ( W e. CLat -> ( ( lub ` W ) ` B ) e. B )
8	4 7	eqeltrd	|- ( W e. CLat -> .1. e. B )