# Elementary Theory – I know that this claim is false. So what's wrong with the proof? If \$ A subseteq B cup C \$, then \$ A subseteq B \$ or \$ A subseteq C \$.

If $$A subseteq B cup C$$, then $$A subseteq B$$ or $$A subseteq C$$, A counter-example to this claim is: $$A = {2,3,4 }$$, $$B = {1,2,3 }$$, $$C = {3,4,5 }$$, But I can not find a mistake in this proof: Suppose $$A subseteq B cup C$$, To let $$x in A$$, Then $$x in B cup C$$, So $$x in B$$ or $$x in C$$, Case 1. $$x in B$$, We have $$x in A rightarrow x in B$$ Thus $$A subseteq B$$, Case 2 $$x in C$$, We have $$x in A rightarrow x in C$$ Thus $$A subseteq C$$, Therefore, $$A subseteq B$$ or $$A subseteq C$$,