Theorem chrelat3i

Description: A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chrelat3.1	|- A e. CH
	chrelat3.2	|- B e. CH
Assertion	chrelat3i	|- ( A C_ B <-> A. x e. HAtoms ( x C_ A -> x C_ B ) )

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	|- A e. CH
2		chrelat3.2	|- B e. CH
3		chrelat3	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> A. x e. HAtoms ( x C_ A -> x C_ B ) ) )
4	1 2 3	mp2an	|- ( A C_ B <-> A. x e. HAtoms ( x C_ A -> x C_ B ) )