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begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] 1. SETS AND ELEMENTS. SUBSETS. "end quote
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Please read these notes from the beginning to here.

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

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