Theorem cvjust

Description: Every set is a class. Proposition 4.9 of TakeutiZaring p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv , which allows us to substitute a setvar variable for a class variable. See also cab and df-clab . Note that this is not a rigorous justification, because cv is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in Jech p. 4 showing that "Every set can be considered to be a class." See abid1 for the version of cvjust extended to classes. (Contributed by NM, 7-Nov-2006) Avoid ax-13 . (Revised by Wolf Lammen, 4-May-2023)

	Ref	Expression
Assertion	cvjust	|- x = { y | y e. x }

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( x = { y | y e. x } <-> A. z ( z e. x <-> z e. { y | y e. x } ) )
2		df-clab	|- ( z e. { y | y e. x } <-> [ z / y ] y e. x )
3		elsb3	|- ( [ z / y ] y e. x <-> z e. x )
4	2 3	bitr2i	|- ( z e. x <-> z e. { y | y e. x } )
5	1 4	mpgbir	|- x = { y | y e. x }