1.8 How to negate a conditional statement

0:00 Introduction
0:10 Statement & Recap
1:04 The scope of x
1:42 Negative the statement
2:09 Essential takeaways

[MathMaths-ex]

Great tutorial.
At 1:18 why do you say, " for all x in real numbers"? Should it just not be " for all x ..." (without 'in real numbers')?
Am asking this because we know what set A is - a subset of real numbers. But we don't know what x is apart from it being an element in set A, so we can't conclude that x is in real numbers.

[raymondliu]

In the negation, why is it "there exists at least one xER s.t. ...." instead of "for all xER ... "?