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Theorem cvjust 2032
 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 1241, which allows us to substitute a setvar variable for a class variable. See also cab 2023 and df-clab 2024. Note that this is not a rigorous justification, because cv 1241 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 2031 . 2 (x = {yy x} ↔ z(z xz {yy x}))
2 df-clab 2024 . . 3 (z {yy x} ↔ [z / y]y x)
3 elsb3 1849 . . 3 ([z / y]y xz x)
42, 3bitr2i 174 . 2 (z xz {yy x})
51, 4mpgbir 1339 1 x = {yy x}
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030 This theorem is referenced by: (None)
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