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Theorem cvjust 2348
 Description: Every setvar 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 1641, which allows us to substitute a setvar variable for a class variable. See also cab 2339 and df-clab 2340. Note that this is not a rigorous justification, because cv 1641 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 = {y y x}
Distinct variable group:   x,y

Proof of Theorem cvjust
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2347 . 2 (x = {y y x} ↔ z(z xz {y y x}))
2 df-clab 2340 . . 3 (z {y y x} ↔ [z / y]y x)
3 elsb3 2103 . . 3 ([z / y]y xz x)
42, 3bitr2i 241 . 2 (z xz {y y x})
51, 4mpgbir 1550 1 x = {y y x}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346 This theorem is referenced by: (None)
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