begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §1. SETS AND ELEMENTS. SUBSETS. "end quote
These pages assume that the reader has the book and is reading along.
topic index     contents

SET EQUALITY IN TERMS OF SUBSETS

We can use ...
begin quote ita-1-1-6-pg3 "
[...] is shorthand for the statement "if   x X,   then   x Y".
"end quote

... to show that ...
begin quote ita-1-1-6-pg3 "
X = Y is equivalent to the two statements     and  .
"end quote

... because we had above the following ...

... and each direction of the double arrow can be read either as the word 'implies' or using the words 'if' (what is at the tail of the arrow is true) 'then' (what is at the head of the arrow is true). Consequently, the condition in parentesis to the right of the first double arrow is the same as the 'two statements' in the second quote above:

 The right hand direction is 'if xX then xY' which is see the first quote above The left hand direction is 'if xY then xX' which is interchange the roles of the upper case X and Y in the first quote above