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Theorem cvjust 2398
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 1406, which allows us to substitute a setvar variable for a class variable. See also cab 2389 and df-clab 2390. Note that this is not a rigorous justification, because cv 1406 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 2397 . 2  |-  ( x  =  { y  |  y  e.  x }  <->  A. z ( z  e.  x  <->  z  e.  {
y  |  y  e.  x } ) )
2 df-clab 2390 . . 3  |-  ( z  e.  { y  |  y  e.  x }  <->  [ z  /  y ] y  e.  x )
3 elsb3 2204 . . 3  |-  ( [ z  /  y ] y  e.  x  <->  z  e.  x )
42, 3bitr2i 252 . 2  |-  ( z  e.  x  <->  z  e.  { y  |  y  e.  x } )
51, 4mpgbir 1645 1  |-  x  =  { y  |  y  e.  x }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    = wceq 1407   [wsb 1765    e. wcel 1844   {cab 2389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396
This theorem is referenced by:  cnambfre  31448
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