Theorem cvjust 2110

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 1313, which allows us to substitute a setvar variable for a class variable. See also cab 2101 and df-clab 2102. Note that this is not a rigorous justification, because cv 1313 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 2109	. 2 |- ( x = { y | y e. x } <-> A. z ( z e. x <-> z e. { y | y e. x } ) )
2		df-clab 2102	. . 3 |- ( z e. { y | y e. x } <-> [ z / y ] y e. x )
3		elsb3 1927	. . 3 |- ( [ z / y ] y e. x <-> z e. x )
4	2, 3	bitr2i 184	. 2 |- ( z e. x <-> z e. { y | y e. x } )
5	1, 4	mpgbir 1412	1 |- x = { y | y e. x }

Syntax hints:   <-> wb 104   = wceq 1314   e. wcel 1463   [wsb 1718   {cab 2101

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108

This theorem is referenced by: (None)