+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; /2/.
+nqed.
+
+nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; /2/.
+nqed.
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+/3/. nqed.
+