A,B,C are sets we know P(A)∪P(B)=P(C) and we need to prove A=C or B=C
my way: we know P(A)∪P(B)=P(C) so P(A)∪P(B)⊆P(C) and P(C)⊆P(A)∪P(B)
P(A)∪P(B)⊆P(C) we know A⊆A and B⊆B so A∈P(A) and B∈P(B) so A∈P(C) and B∈P(C) so A⊆C and B⊆C
P(C)⊆P(A)∪P(B) the same way we get: C⊆A and C⊆B
and then we get A=C and B=C which is different (but still proofs) from what we need to proof. Am I wrong anywhere? thx!
Most of it looks good to me. However, the statement $C \subseteq A$ and $C \subseteq B$ isn't true. Try working through that section explicitly and you'll see that you actually get $C \subseteq A$ or $C \subseteq B$, giving the desired result $A = C$ or $B = C$.