Question: I have noticed a kind of contradiction arising due to set-builder form involving universal set$(U)$ while I was performing operations on set theory. 

Here it is -

> 
> Let us consider an arbitrary non-empty set $B$.
>  Then, $B = \{x : x \in B\} = \{x : x \in B \wedge x \in U\} = \{x : x \in B \cap U\} = \{x : x \in U\} = U$
> 
> 
> 

But this is contradiction as $B \not= U$.
Instead $B \subseteq U$.

Is it really a contradiction or my misconception ?

A detailed explanation would be helpful.

Answer: 
As has been explained in the comments *(and if one of the original commenters adds an answer, I'll delete this one)*, the error is not a subtle logical issue but rather an intersection calculation.

You ask,

> 
> Why can't we consider $B\cap U=U$ instead of $B\cap U=B$?
> 
> 
> 

Well, the answer is: the first one is **not necessarily true** given our hypotheses on $B$ and $U$, while the second one is **necessarily true** given those hypotheses. I suspect the special character of $U$ (= the universal set) is making this more confusing than it should be. 

Forgetting about universal sets, remember that whenever $X\subseteq Y$ we have $X\cap Y=X$, but we do **not** necessarily have $X\cap Y=Y$ (indeed, we won't unless $X=Y$). For example, take $X=\{1\}, Y=\{1,2\}$; we have $$\{1\}\cap \{1,2\}=\{1\}\quad\mbox{but}\quad\{1\}\cap\{1,2\}\not=\{1,2\}.$$

---

Of course the universal set *does* pose [serious issues](https://plato.stanford.edu/entries/russell-paradox/), and [the usual choice of axioms for set theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) in fact prohibits such an object *(although some [other set theories](https://en.wikipedia.org/wiki/New_Foundations) allow it at the cost of weakening other axioms)*; but that's irrelevant to the particular issue here, which is really just an issue with subsets and intersections.
The mistake might originate from a lack of familiarity with the propositional constant "***t***". 

The sentence " x belongs to U " is a tautology , whatever x might be ( since, the defintion of the universal set means that any element you are considering belongs to the universal set). 

Now, in propositional logic there is a propositional constant denoted by " ***t***" which is defined as : " truth" , "the tautology" , the proposition that is equivalent to any tautology. 

Remark : there is also a propositional constant called : " ***f***" " false" ( and that is defined as : the proposition that is equivalent to any antilogy. 

Now, propositional logic tells you that, for any proposition P : 

(P& ***t***) is equivalent to P. 

This is " identity law for tautologies" 
( <http://mathonline.wikidot.com/the-laws-of-propositional-logic> , see also Lipschutz, Schaum's outline of set theory, at *archive.org*) 

Explanation : in order a conjunction to be true, both propositions have to be true; since ***t*** is true in all possible case ( being a tautology), everything depends on the truth value of P: if P is true the whole conjunction in true, if P is false, the whole conjunction is false. 

With this in mind, what is the set : B Inter U ? 

B Inter U = { x | x belongs to B & x belongs to U } 

But, as we said, the sentence " x belongs to U " is equivalent to ***t*** . 

So, 

B Inter U 

= { x | x belongs to B & x belongs to U } 

= { x | x belongs to B & ***t*** }

= { x | x belongs to B}

= B