Theorem lubprdm

Description: The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024)

	Ref	Expression
Hypotheses	lubpr.k	|- ( ph -> K e. Poset )
	lubpr.b	|- B = ( Base ` K )
	lubpr.x	|- ( ph -> X e. B )
	lubpr.y	|- ( ph -> Y e. B )
	lubpr.l	|- .<_ = ( le ` K )
	lubpr.c	|- ( ph -> X .<_ Y )
	lubpr.s	|- ( ph -> S = { X , Y } )
	lubpr.u	|- U = ( lub ` K )
Assertion	lubprdm	|- ( ph -> S e. dom U )

Proof

Step	Hyp	Ref	Expression
1		lubpr.k	|- ( ph -> K e. Poset )
2		lubpr.b	|- B = ( Base ` K )
3		lubpr.x	|- ( ph -> X e. B )
4		lubpr.y	|- ( ph -> Y e. B )
5		lubpr.l	|- .<_ = ( le ` K )
6		lubpr.c	|- ( ph -> X .<_ Y )
7		lubpr.s	|- ( ph -> S = { X , Y } )
8		lubpr.u	|- U = ( lub ` K )
9	1 2 3 4 5 6 7 8	lubprlem	|- ( ph -> ( S e. dom U /\ ( U ` S ) = Y ) )
10	9	simpld	|- ( ph -> S e. dom U )