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	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chlejb1i	$$⊢A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		ssid	$$⊢B\subseteq B$$
4	1 2 2	chlubii	$$⊢\left(A\subseteq B\wedge B\subseteq B\right)\to A{\vee }_{\mathscr{H}}B\subseteq B$$
5	3 4	mpan2	$$⊢A\subseteq B\to A{\vee }_{\mathscr{H}}B\subseteq B$$
6	2 1	chub2i	$$⊢B\subseteq A{\vee }_{\mathscr{H}}B$$
7	5 6	jctir	$$⊢A\subseteq B\to \left(A{\vee }_{\mathscr{H}}B\subseteq B\wedge B\subseteq A{\vee }_{\mathscr{H}}B\right)$$
8		eqss	$$⊢A{\vee }_{\mathscr{H}}B=B\leftrightarrow \left(A{\vee }_{\mathscr{H}}B\subseteq B\wedge B\subseteq A{\vee }_{\mathscr{H}}B\right)$$
9	7 8	sylibr	$$⊢A\subseteq B\to A{\vee }_{\mathscr{H}}B=B$$
10	1 2	chub1i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$
11		eqimss	$$⊢A{\vee }_{\mathscr{H}}B=B\to A{\vee }_{\mathscr{H}}B\subseteq B$$
12	10 11	sstrid	$$⊢A{\vee }_{\mathscr{H}}B=B\to A\subseteq B$$
13	9 12	impbii	$$⊢A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B$$