Theorem chlejb2

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlejb2	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow B{\vee }_{\mathscr{H}}A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		chlejb1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B\right)$$
2		chjcom	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}A$$
3	2	eqeq1d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{\vee }_{\mathscr{H}}B=B\leftrightarrow B{\vee }_{\mathscr{H}}A=B\right)$$
4	1 3	bitrd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow B{\vee }_{\mathscr{H}}A=B\right)$$