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Theorem cvjust 2051
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 1258, which allows us to substitute a setvar variable for a class variable. See also cab 2042 and df-clab 2043. Note that this is not a rigorous justification, because cv 1258 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 2050 . 2  |-  ( x  =  { y  |  y  e.  x }  <->  A. z ( z  e.  x  <->  z  e.  {
y  |  y  e.  x } ) )
2 df-clab 2043 . . 3  |-  ( z  e.  { y  |  y  e.  x }  <->  [ z  /  y ] y  e.  x )
3 elsb3 1868 . . 3  |-  ( [ z  /  y ] y  e.  x  <->  z  e.  x )
42, 3bitr2i 178 . 2  |-  ( z  e.  x  <->  z  e.  { y  |  y  e.  x } )
51, 4mpgbir 1358 1  |-  x  =  { y  |  y  e.  x }
Colors of variables: wff set class
Syntax hints:    <-> wb 102    = wceq 1259    e. wcel 1409   [wsb 1661   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by: (None)
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