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Theorem cvjust 2253
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 1618, which allows us to substitute a set variable for a class variable. See also cab 2244 and df-clab 2245. Note that this is not a rigorous justification, because cv 1618 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
StepHypRef Expression
1 dfcleq 2252 . 2  |-  ( x  =  { y  |  y  e.  x }  <->  A. z ( z  e.  x  <->  z  e.  {
y  |  y  e.  x } ) )
2 df-clab 2245 . . 3  |-  ( z  e.  { y  |  y  e.  x }  <->  [ z  /  y ] y  e.  x )
3 elsb3 2066 . . 3  |-  ( [ z  /  y ] y  e.  x  <->  z  e.  x )
42, 3bitr2i 243 . 2  |-  ( z  e.  x  <->  z  e.  { y  |  y  e.  x } )
51, 4mpgbir 1544 1  |-  x  =  { y  |  y  e.  x }
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   [wsb 1883   {cab 2244
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251
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