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- Is ##A \subset B \iff \exists x \in B \land x \not\in A## correct?

N.B. I ommitted the other half of the statement for emphasis, see the complete statement below.

##A \subset B## means that ##\exists x \in B ## such that ##x \not\in A##. Is this logically equivalent to ##\exists x \in B \land x \not\in A##? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$

I have tried consulting https://math.stackexchange.com/questions/432345/such-that-logical-symbol, it seemes that there is no logical notation for "such that". I would thus like to confirm whether the above statement is correct (I mean whether it fulfils the notational convention).

Also, is this statement true and equivalent to the statement above: $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall x \in A \land x\in B)}$$ I understand that it is bound to cause confusion, since I used "x" in two different, concurrent statements, but is the "x" only circumscribed to each of the simple statements that form the compound statement, or is it somewhat like a global variable, such that I should use a different variable (in this case y) for different statements? Thanks in advance!

I have tried consulting https://math.stackexchange.com/questions/432345/such-that-logical-symbol, it seemes that there is no logical notation for "such that". I would thus like to confirm whether the above statement is correct (I mean whether it fulfils the notational convention).

Also, is this statement true and equivalent to the statement above: $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall x \in A \land x\in B)}$$ I understand that it is bound to cause confusion, since I used "x" in two different, concurrent statements, but is the "x" only circumscribed to each of the simple statements that form the compound statement, or is it somewhat like a global variable, such that I should use a different variable (in this case y) for different statements? Thanks in advance!