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Theorem cvjust 2018
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 1227, which allows us to substitute a setvar variable for a class variable. See also cab 2009 and df-clab 2010. Note that this is not a rigorous justification, because cv 1227 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 = {yy x}
Distinct variable group:   x,y

Proof of Theorem cvjust
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2017 . 2 (x = {yy x} ↔ z(z xz {yy x}))
2 df-clab 2010 . . 3 (z {yy x} ↔ [z / y]y x)
3 elsb3 1835 . . 3 ([z / y]y xz x)
42, 3bitr2i 174 . 2 (z xz {yy x})
51, 4mpgbir 1322 1 x = {yy x}
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228   wcel 1375  [wsb 1628  {cab 2009
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016
This theorem is referenced by: (None)
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