Im trying to prove the following The second part especially

I\'m trying to prove the following (The second part especially):

1.) A B A

For this one, here is what I have so far (I feel like from this step I can conclude the proof?)

2.) A B iff A B = A

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Solution

Proof.

First assume that A B. If x A B, then x A and x B by definition, so in particular x A. This proves A B A. Now if x A, then by assumption x B, too, so x A B. This proves A A B. Together this implies A = A B. Now assume that A B = A. If x A, then by assumption x A B, so x A and x B. In particular, x B. This proves A B.