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Theorem cvjust 2396
 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 1648, which allows us to substitute a set variable for a class variable. See also cab 2387 and df-clab 2388. Note that this is not a rigorous justification, because cv 1648 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 2395 . 2
2 df-clab 2388 . . 3
3 elsb3 2150 . . 3
42, 3bitr2i 242 . 2
51, 4mpgbir 1556 1
 Colors of variables: wff set class Syntax hints:   wb 177   wceq 1649  wsb 1655   wcel 1721  cab 2387 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2388  df-cleq 2394
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