Q:

Question on associative property of sets

In the proof of

A ∩ (B - C) = (A ∩ B) - (A ∩ C)

Most of the proofs I have seen involve the following steps:

i) Let x be an arbitrary element of A ∩ (B - C)
ii) ⇒ x ∈ A and x ∈ (B - C)
iii) ⇒ x ∈ A and x ∈ B, but x ∉ C
iv) ⇒ x ∈ (A ∩ B), but x ∉ (A ∩ C)
The proof is continued further.

I have problem in Part iii and iv. How did we reach part iv from iii. According to me we should have used the associative law that A ∩ (B ∩ C)=(A ∩ B) ∩ C .
According to me the following steps should be taken

x ∈ A and x ∈ B, but x ∉ C
x ∈ A and x ∈ B, and x ∉ C
x ∈ (A ∩ B) and x ∉ C
x ∈ (A ∩ B) and x ∈ C'
x ∈ (A ∩ B)∩ C'
x ∈ (A ∩ B)- C
This leads to nowhere

Please tell me why is part iv the correct step after iii and not what I suppose.
Thank you

A:

Direct set algebra proof.  
$A \cap (B - C) = A \cap B \cap C^c =  $
$(A \cap B \cap A^c) \cup (A \cap B \cap C^c) =  
A \cap B \cap (A^c \cup C^c) =$
$A \cap B \cap (A \cap C)^c = A \cap B - (A \cap C)
$

