Theorem cvjust 2035

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 1242, which allows us to substitute a setvar variable for a class variable. See also cab 2026 and df-clab 2027. Note that this is not a rigorous justification, because cv 1242 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." (Contributed by NM, 7-Nov-2006.)

Assertion

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

Distinct variable group:   x, y

Proof of Theorem cvjust

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2034	. 2 |- ( x = { y | y e. x } <-> A. z ( z e. x <-> z e. { y | y e. x } ) )
2		df-clab 2027	. . 3 |- ( z e. { y | y e. x } <-> [ z / y ] y e. x )
3		elsb3 1852	. . 3 |- ( [ z / y ] y e. x <-> z e. x )
4	2, 3	bitr2i 174	. 2 |- ( z e. x <-> z e. { y | y e. x } )
5	1, 4	mpgbir 1342	1 |- x = { y | y e. x }

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   [wsb 1645   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033

This theorem is referenced by: (None)