Theorem cvjust 2617

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 1482, which allows us to substitute a setvar variable for a class variable. See also cab 2608 and df-clab 2609. Note that this is not a rigorous justification, because cv 1482 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 2616	. 2 |- ( x = { y | y e. x } <-> A. z ( z e. x <-> z e. { y | y e. x } ) )
2		df-clab 2609	. . 3 |- ( z e. { y | y e. x } <-> [ z / y ] y e. x )
3		elsb3 2434	. . 3 |- ( [ z / y ] y e. x <-> z e. x )
4	2, 3	bitr2i 265	. 2 |- ( z e. x <-> z e. { y | y e. x } )
5	1, 4	mpgbir 1726	1 |- x = { y | y e. x }

Syntax hints:   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615

This theorem is referenced by:  cnambfre  33458