Theorem chrelat4i

Description: A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chrelat3.1	⊢ 𝐴 ∈ Cℋ
	chrelat3.2	⊢ 𝐵 ∈ Cℋ
Assertion	chrelat4i	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	⊢ 𝐴 ∈ Cℋ
2		chrelat3.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chrelat3i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) )
4	2 1	chrelat3i	⊢ ( 𝐵 ⊆ 𝐴 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) )
5	3 4	anbi12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) ↔ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ∧ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) ) )
6		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
7		ralbiim	⊢ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) ↔ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ∧ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) ) )
8	5 6 7	3bitr4i	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) )