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Metamath Proof Explorer

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 ) )