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Theorem cvjust 2135
 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 1331, which allows us to substitute a setvar variable for a class variable. See also cab 2126 and df-clab 2127. Note that this is not a rigorous justification, because cv 1331 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 2134 . 2
2 df-clab 2127 . . 3
3 elsb3 1952 . . 3
42, 3bitr2i 184 . 2
51, 4mpgbir 1430 1
 Colors of variables: wff set class Syntax hints:   wb 104   wceq 1332   wcel 1481  wsb 1736  cab 2126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133 This theorem is referenced by: (None)
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