Theorem latasymd

Description: Deduce equality from lattice ordering. ( eqssd analog.) (Contributed by NM, 18-Nov-2011)

	Ref	Expression
Hypotheses	latasymd.b	|- B = ( Base ` K )
	latasymd.l	|- .<_ = ( le ` K )
	latasymd.3	|- ( ph -> K e. Lat )
	latasymd.4	|- ( ph -> X e. B )
	latasymd.5	|- ( ph -> Y e. B )
	latasymd.6	|- ( ph -> X .<_ Y )
	latasymd.7	|- ( ph -> Y .<_ X )
Assertion	latasymd	|- ( ph -> X = Y )

Proof

Step	Hyp	Ref	Expression
1		latasymd.b	|- B = ( Base ` K )
2		latasymd.l	|- .<_ = ( le ` K )
3		latasymd.3	|- ( ph -> K e. Lat )
4		latasymd.4	|- ( ph -> X e. B )
5		latasymd.5	|- ( ph -> Y e. B )
6		latasymd.6	|- ( ph -> X .<_ Y )
7		latasymd.7	|- ( ph -> Y .<_ X )
8	1 2	latasymb	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
9	3 4 5 8	syl3anc	|- ( ph -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
10	6 7 9	mpbi2and	|- ( ph -> X = Y )