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Theorem cvjust 2280
Description: Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1624, which allows us to substitute a set variable for a class variable. See also cab 2271 and df-clab 2272. Note that this is not a rigorous justification, because cv 1624 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.
StepHypRef Expression
1 dfcleq 2279 . 2  |-  ( x  =  { y  |  y  e.  x }  <->  A. z ( z  e.  x  <->  z  e.  {
y  |  y  e.  x } ) )
2 df-clab 2272 . . 3  |-  ( z  e.  { y  |  y  e.  x }  <->  [ z  /  y ] y  e.  x )
3 elsb3 2044 . . 3  |-  ( [ z  /  y ] y  e.  x  <->  z  e.  x )
42, 3bitr2i 241 . 2  |-  ( z  e.  x  <->  z  e.  { y  |  y  e.  x } )
51, 4mpgbir 1539 1  |-  x  =  { y  |  y  e.  x }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1625   [wsb 1631    e. wcel 1686   {cab 2271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278
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