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Theorem cvjust 2248
 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 2239 and df-clab 2240. 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
Distinct variable group:   ,

Proof of Theorem cvjust
StepHypRef Expression
1 dfcleq 2247 . 2
2 df-clab 2240 . . 3
3 elsb3 2063 . . 3
42, 3bitr2i 243 . 2
51, 4mpgbir 1544 1
 Colors of variables: wff set class Syntax hints:   wb 178   wceq 1619   wcel 1621  wsb 1882  cab 2239 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 2234 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 1883  df-clab 2240  df-cleq 2246
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