Metamath Proof Explorer

Theorem sltssep

Description: The separation property of surreal set less-than. (Contributed by Scott Fenton, 8-Dec-2021)

Ref Expression
Assertion sltssep ⊢ A ≪ s B → ∀ x ∈ A ∀ y ∈ B x < s y

Proof

Step Hyp Ref Expression
1 brslts ⊢ A ≪ s B ↔ A ∈ V ∧ B ∈ V ∧ A ⊆ No ∧ B ⊆ No ∧ ∀ x ∈ A ∀ y ∈ B x < s y
2 simpr3 ⊢ A ∈ V ∧ B ∈ V ∧ A ⊆ No ∧ B ⊆ No ∧ ∀ x ∈ A ∀ y ∈ B x < s y → ∀ x ∈ A ∀ y ∈ B x < s y
3 1 2 sylbi ⊢ A ≪ s B → ∀ x ∈ A ∀ y ∈ B x < s y