I need to prove that $$A\subseteq B \implies A\cup B=B$$ I defined the subset relation as the statement $x\in A\Rightarrow x\in B$. I tried to convert the claim into a logic statement, then proceeded to simplify the statement $$(x\in A\Rightarrow x\in B)\Rightarrow(x\in A\vee x\in B),$$ and I simplified it to $x\in A\vee x\in B$. Only after I did this did I realize that this didn't really prove anything.
This fact seems to be taken axiomatically—I haven't been able to find any proofs for it.
I found that it is proposed under Wikipedia's Proposition 8 under ‘Algebra of Sets’ that this is true, and equivalent to the subset relation. A proof of this, however, escapes me.
I also realize that it would be sufficient to be able to convert the antecedent to the consequent or vice versa (which would result in the tautology $p\Rightarrow p$ and thus prove the statement), though I do not know how to do this.