Variables (X : {set T}) (P : {set {set T}}) (D : {set T}).
Hypothesis trPX : is_transversal X P D.

Lemma transversal_sub : X \subset D. Proof. by case/and3P: trPX. Qed.
Let tiP : trivIset P. Proof. by case/andP: trPX => /and3P[]. Qed.