Theorem chlejb1i

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	|- A e. CH
	chjcl.2	|- B e. CH
Assertion	chlejb1i	|- ( A C_ B <-> ( A vH B ) = B )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3		ssid	|- B C_ B
4	1 2 2	chlubii	|- ( ( A C_ B /\ B C_ B ) -> ( A vH B ) C_ B )
5	3 4	mpan2	|- ( A C_ B -> ( A vH B ) C_ B )
6	2 1	chub2i	|- B C_ ( A vH B )
7	5 6	jctir	|- ( A C_ B -> ( ( A vH B ) C_ B /\ B C_ ( A vH B ) ) )
8		eqss	|- ( ( A vH B ) = B <-> ( ( A vH B ) C_ B /\ B C_ ( A vH B ) ) )
9	7 8	sylibr	|- ( A C_ B -> ( A vH B ) = B )
10	1 2	chub1i	|- A C_ ( A vH B )
11		eqimss	|- ( ( A vH B ) = B -> ( A vH B ) C_ B )
12	10 11	sstrid	|- ( ( A vH B ) = B -> A C_ B )
13	9 12	impbii	|- ( A C_ B <-> ( A vH B ) = B )