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Theorem cvjust 2013
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 1225, which allows us to substitute a setvar variable for a class variable. See also cab 2004 and df-clab 2005. Note that this is not a rigorous justification, because cv 1225 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  {  |  }
Distinct variable group:   ,

Proof of Theorem cvjust
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2012 . 2  {  |  }  {  |  }
2 df-clab 2005 . . 3  {  |  }
3 elsb3 1830 . . 3
42, 3bitr2i 174 . 2 
{  |  }
51, 4mpgbir 1318 1  {  |  }
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1226   wcel 1370  wsb 1623   {cab 2004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011
This theorem is referenced by: (None)
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