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 $,