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 ∈ Cℋ
	chrelat3.2	⊢ 𝐵 ∈ Cℋ
Assertion	chrelat3i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	⊢ 𝐴 ∈ Cℋ
2		chrelat3.2	⊢ 𝐵 ∈ Cℋ
3		chrelat3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) )