If I let $A=\emptyset$,$B=C=\{x\}$, do I have a valid counterexample?
2026-05-05 21:18:11.1778015891
disprove: $\forall A,B,C(A\cup B=A\cup C \Rightarrow B=C)$
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Simple counterexample $B\subset A$ and $C\subset A$. Then $A=A\cup B=A\cup C$. There is no other relationship between $B$ and $C$. Simple choice, make $C=A-B$.