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 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	$$⊢x=\left\{y|y\in x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢x=\left\{y|y\in x\right\}\leftrightarrow \forall z\left(z\in x\leftrightarrow z\in \left\{y|y\in x\right\}\right)$$
2		df-clab	$$⊢z\in \left\{y|y\in x\right\}\leftrightarrow \left[z/y\right]y\in x$$
3		elsb1	$$⊢\left[z/y\right]y\in x\leftrightarrow z\in x$$
4	2 3	bitr2i	$$⊢z\in x\leftrightarrow z\in \left\{y|y\in x\right\}$$
5	1 4	mpgbir	$$⊢x=\left\{y|y\in x\right\}$$