Let $$A, B, C$$ and $$D$$ be four non-empty sets. The contrapositive statement of "If $$A \subseteq B$$ and $$B \subseteq D$$, then $$A \subseteq C$$" is

We begin with the given conditional statement:

$$\text{If }A \subseteq B\text{ and }B \subseteq D,\text{ then }A \subseteq C.$$

This has the usual “If-then” logical structure. For clarity, let us assign symbols to the two parts of the statement:

$$P : A \subseteq B \text{ and } B \subseteq D,$$

$$Q : A \subseteq C.$$

So the original statement is of the form $$P \rightarrow Q.$$

Now we recall the logical rule for the contrapositive. For any implication $$P \rightarrow Q,$$ the contrapositive is obtained by negating both parts and reversing the direction:

$$P \rightarrow Q \;\; \text{is equivalent to} \;\; \lnot Q \rightarrow \lnot P.$$

Applying this rule, we first negate $$Q$$:

$$Q = A \subseteq C \quad\Longrightarrow\quad \lnot Q = A \nsubseteq C.$$

Next, we negate $$P$$. Since $$P$$ itself is a conjunction, we must use De Morgan’s law:

$$P = (A \subseteq B) \land (B \subseteq D).$$

According to De Morgan’s law, the negation of a conjunction is the disjunction of the negations:

$$\lnot P = \lnot\big[(A \subseteq B) \land (B \subseteq D)\big]$$

$$\phantom{\lnot P} = (A \nsubseteq B) \lor (B \nsubseteq D).$$

Combining these two results into the contrapositive form $$\lnot Q \rightarrow \lnot P,$$ we obtain:

$$\text{If }A \nsubseteq C,\text{ then }(A \nsubseteq B)\text{ or }(B \nsubseteq D).$$

Comparing this with the options given, we see it matches exactly with Option D.

Hence, the correct answer is Option D.