Theorem cvjust 2224

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 1394, which allows us to substitute a setvar variable for a class variable. See also cab 2215 and df-clab 2216. Note that this is not a rigorous justification, because cv 1394 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 2223	. 2 |- ( x = { y | y e. x } <-> A. z ( z e. x <-> z e. { y | y e. x } ) )
2		df-clab 2216	. . 3 |- ( z e. { y | y e. x } <-> [ z / y ] y e. x )
3		elsb1 2207	. . 3 |- ( [ z / y ] y e. x <-> z e. x )
4	2, 3	bitr2i 185	. 2 |- ( z e. x <-> z e. { y | y e. x } )
5	1, 4	mpgbir 1499	1 |- x = { y | y e. x }

Syntax hints:   <-> wb 105   = wceq 1395   [wsb 1808   e. wcel 2200   {cab 2215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222

This theorem is referenced by: (None)