# Metamath Proof Explorer

## 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 ${⊢}{x}=\left\{{y}|{y}\in {x}\right\}$

### Proof

Step Hyp Ref Expression
1 dfcleq ${⊢}{x}=\left\{{y}|{y}\in {x}\right\}↔\forall {z}\phantom{\rule{.4em}{0ex}}\left({z}\in {x}↔{z}\in \left\{{y}|{y}\in {x}\right\}\right)$
2 df-clab ${⊢}{z}\in \left\{{y}|{y}\in {x}\right\}↔\left[{z}/{y}\right]{y}\in {x}$
3 elsb3 ${⊢}\left[{z}/{y}\right]{y}\in {x}↔{z}\in {x}$
4 2 3 bitr2i ${⊢}{z}\in {x}↔{z}\in \left\{{y}|{y}\in {x}\right\}$
5 1 4 mpgbir ${⊢}{x}=\left\{{y}|{y}\in {x}\right\}$