Metamath Proof Explorer

Theorem ssltsep

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

Ref Expression
Assertion ssltsep ⊢A≪sB→∀x∈A∀y∈Bx<sy

Proof

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