Greetings. I have been studying David Poole's Linear Algebra textbook and I discovered in the Appendix that if it can be shown that if the set A is a sub-set of B and B is a sub-set of A, then the set A is exactly the same as the set B. And this all seems intuitively plausible, but for the life of me I couldn't prove it. No proof was adduced in the textbook, but I assume such a proof exists. Is it simply an axiom or is it derived from something else? I tried to take it along these lines: Some of the elements of B form the entirety of the elements of A. And the some of the elements of B form the entirety of the elements of B. I even tried to pictorialize it with an ad hoc variation of the Venn diagram, where the bubble representing B curled around and entered A in a sort of two-dimensional Klein bottle. No help here either. Any suggestions?