Where did the term "set-builder notation" come from?

Question

In math stack exchange  I often see notations like ${xinmathbb Q:x^2<2}$ being called instances of  set builder notation.  When I went to school we (that is, I, my fellow students, my teachers, and authors of textbooks such as P. Halmos, N. Bourbaki, et al.) used such notations all the time but never with that description.

My question is, when was this name introduced?  Are the references given in the Wikipedia article the original introducers, or just modern users of a recent term for an older idea?

Added, after comments and after Conifold's superb answer goaded me to look a tiny bit harder:

A 1948 University of Chicago Press publication Fundamental Mathematics Volume 1 Prepared for the General Course 1 in the College ("by the College mathematics staff" with a list of 13 names)  uses the term "SET-BUILDER" on p.25:

Where $W$ is a property, we use the expression "$S_x[W(x)]$"
  to stand for "the set of all objects $x$ which have the property $W$"
  or "the set of all objects $x$ for which '$W(x)$' is true'' or "the
  set of all $x$ such that $W(x)$". The sign $S$ is known as the SET-BUILDER.

It is photolithographed from a typescript original.  What I render as $S_x$ has the $x$ directly under the $S$.

A similar Chicago 1954 book, Concepts and structure of mathematics, by the College mathematics staff, uses the term "set-builder notation", but Google Books doesn't give enough context to show that it is exactly the same as the 2019 sense of the term.

Conifold · Accepted Answer

The nickname appears to be a creation of the New Math movement, and spread from the math education literature.

The notation itself in its modern form can be traced back to Lefschetz's Algebraic Topology (1942), and variants appear already in Principia (1910) and von Neumann's Zur Einführung der transfiniten Zahlen (1923). See Who first discovered the concept corresponding to the symbol of class comprehension? for many more details. However, mathematicians did not use the name "set-builder". Bernays (1958) calls it "class operator" and Suppes (1960) "definition by abstraction". The name does not appear before 1957, but in 1958 we find it in the lively discussions of the high school curriculum in The Mathematics Teacher. E.g. Rourke's Some implications of twentieth century mathematics for high schools explains:

"We have a convenient notation for de noting solution sets, using the set-builder :
   ${xmid }$. The braces "${ }$" are read "set"; the vertical "$|$" is read "such that." We put the variable on the left-hand side of the vertical bar, and the sentence on the  right-hand side."

And Duren's The maneuvers in set thinking goes deep into the pedagogy:

"We do not have any system of individual names for sets  like the decimal representations of the real numbers. Hence we have no way of  giving the name of a particular set which  is the "answer" to a problem except by such indirect devices as the set-builder: ${x e X | A & B & C}$ = "The set of all elements in X having properties A and B and C"."

The student journal Pi Mu Epsilon still uses scare quotes around set-builder when reviewing Suppes's textbook in 1960.

What happened in 1957 is that the Soviet Union launched an orbital satellite, and a period of existential anxiety in the Western countries known as the Sputnik crisis. One of the responses was to pack high school curriculum with symbolic logic, matrices and sets, among other things, to "catch up" to the Soviet advances (in fairness, some reforms date back to he University of Illinois Committee on School Mathematics from 1951 on). Ironically, while Soviet high school mathematics was reformed in the 1930s, it was not this way. Nor is there a Russian analog of the "set-builder" nickname, according to Russian Wikipedia. After 1958, the New Math, and "set-builder", rapidly spread into textbooks. The earliest I found is the teacher's edition of Mathematics for High School, p.16 (1959):

"The braces ${ }$ used to enclose the elements of a set call
  attention to the fact that we are to think of the collection as
  a singl entity. The set-builder notation ${x: x..... }$ is a
  useful way to represent a set which is characterized by some rule or property, nothing more. In some treatments of the subject vertical bar is used in place of the colon in the set-builder notation. We prefer the colon for typographical reasons."

New Math was controversial from the start, and heavy criticism pushed most of it out of high schools by the end of 1960s. But the nickname stuck.

Where did the term "set-builder notation" come from?

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