Prove that for any sets A,B,C: If A is a proper subset of B, then (C-B) is a proper subset of (C-A)

### Problem Statement:
We are asked to prove that if \( A \) is a proper subset of \( B \), then \( C - B \) is a proper subset of \( C - A \) for any sets \( A \), \( B \), and \( C \).

### Definitions:
1. **Proper Subset**: Set \( A \) is a proper subset of \( B \) (denoted \( A \subsetneq B \)) if:
   - \( A \subseteq B \) (i.e., every element of \( A \) is also in \( B \)),
   - and \( A \neq B \) (i.e., there exists at least one element in \( B \) that is not in \( A \)).
   
2. **Set Difference**: The set difference \( C - X \) is defined as the set of elements in \( C \) that are not in \( X \). Formally, \( C - X = \{ x \in C \mid x \notin X \} \).

### Goal:
To prove that:
\[
A \subsetneq B \implies C - B \subsetneq C - A
\]
which means:
- \( C - B \subseteq C - A \) (subset condition),
- \( C - B \neq C - A \) (proper subset condition).

### Proof:

#### 1. Subset Condition: \( C - B \subseteq C - A \)
Let \( x \in C - B \). By definition of set difference, this means:
\[
x \in C \quad \text{and} \quad x \notin B.
\]
Since \( A \subseteq B \), we know that every element of \( A \) is also in \( B \), so:
\[
x \notin B \implies x \notin A.
\]
Thus, \( x \in C - A \). Therefore, we have shown that:
\[
x \in C - B \implies x \in C - A,
\]
which proves that:
\[
C - B \subseteq C - A.
\]

#### 2. Proper Subset Condition: \( C - B \neq C - A \)
Since \( A \subsetneq B \), there exists some element \( y \in B \) such that \( y \notin A \). Now, consider the case where \( y \in C \). Since \( y \in B \) and \( y \notin A \), we have:
\[
y \notin C - B \quad \text{(because } y \in B\text{)}.
\]
However, since \( y \notin A \) and \( y \in C \), we have:
\[
y \in C - A.
\]
Thus, there exists an element \( y \in C - A \) that is not in \( C - B \), meaning \( C - B \neq C - A \).

### Conclusion:
We have shown both that:
1. \( C - B \subseteq C - A \),
2. \( C - B \neq C - A \).

Therefore, \( C - B \) is a proper subset of \( C - A \), and the statement is proved.

### Do you have any questions or would you like further details on any part of this proof?

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Here are 5 related questions you might want to explore:
1. What is the difference between a subset and a proper subset?
2. Can we generalize this result to infinite sets?
3. How does this result change if \( A = B \)?
4. What is the relationship between set difference and set union?
5. How does the result change if \( A \) is not a subset of \( B \)?

**Tip:** Always double-check the definitions of key terms like "subset" and "proper subset" when working with proofs involving sets. Misunderstandings in definitions can lead to incorrect conclusions.

Prove that for any sets A, B, C: If A is a proper subset of B, then (C - B) is a proper subset of (C - A)

Explore the proof that for any sets A, B, C: If A is a proper subset of B, then (C - B) is a proper subset of (C - A). This proof covers set theory concepts such as proper subsets and set difference, with a detailed step-by-step explanation.

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