Two Properties in the Elementary Set Theory

My first post is about two properties in the elementary set theory. You can see an amazing proof for them from me in the following.

(A\cup C)\cap(B\cup C^c)\subseteq A\cup B

A\cap B\subseteq(A\cap C)\cup(B\cap C^c)

In proof I used the following property in logic

(P\Rightarrow Q)\land(R\Rightarrow S) \vdash (P\land R)\Rightarrow( Q\land S)

that is called “constructive dilemma”.

First Property:

Let x\in(A\cup C)\cap(B\cup C^c), then I can write

(x\in A\lor x\in C)\land( x\in B\lor x\in C^c)

(x\in A^c\Rightarrow x\in C)\land(x\in B^c\Rightarrow x\in C^c)

(x\in A^c\land x\in B^c)\Rightarrow( x\in C\land x\in C^c)

\lnot(x\in A^c\land x\in B^c)\lor(x\in C\land x\in C^c)

(x\in A\lor x\in B)\lor(x\in C\land x\in C^c)

which implies that x\in A\cup B.

Second Property:

Let x\in A\cap B, then

(x\in A\land x\in B)\land\lnot(x\in C^c\land x\in C)

\lnot \left((x\in A\land x\in B)\Rightarrow( x\in C^c\land x\in C)\right)

\lnot \left((x\in A\Rightarrow x\in C^c)\land(x\in B\Rightarrow x\in C)\right)

\lnot\left((x\in A^c\lor x\in C^c)\land(x\in B^c\lor x\in C)\right)

(x\in A\land x\in C)\lor(x\in B\land x\in C^c)

which implies that x\in (A\cap C)\cup(B\cap C^c).

Two Properties in the Elementary Set Theory