no goals
R : Type u
S✝ : Type v
T : Type w
inst✝³ : NonUnitalNonAssocSemiring R
M : Subsemigroup R
inst✝² : NonUnitalNonAssocSemiring S✝
inst✝¹ : NonUnitalNonAssocSemiring T
F : Type ?u.284510
inst✝ : NonUnitalRingHomClass F R S✝
S : Set (NonUnitalSubsemiring R)
Sne : Set.Nonempty S
hS : DirectedOn (fun x x_1 => x ≤ x_1) S
x : R
⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ (s : NonUnitalSubsemiring R) (_ : s ∈ S), ↑s
simp [mem_sSup_of_directedOn Sne hS]