If by "contain x" you mean "has x as a subset" then what you say is false (i give an example below). On the ohter hand, if you mean "has x as an element", then i don't see why you would expect this to happen in order to be equal to x. In fact, x itself cannot ever contain x as an element; yet x=x.
Here's an example of the definition:
If 0 denotes the emptyset, consider
S = {0, {0}, {0,{0}}}
with the ordering relation x<y being "x is a proper subset of y".
Then if x is an element of S, either
1) x=0
2) x={0},
or
3) x={0,{0}}.
In either case, you can easily check that the set {z in S: z<x} is exactly x. So in fact S is an ordinal (usuall known as 2 :)
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