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Theorem cvjust 2615
 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 1480, which allows us to substitute a setvar variable for a class variable. See also cab 2606 and df-clab 2607. Note that this is not a rigorous justification, because cv 1480 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 = {𝑦𝑦𝑥}
Distinct variable group:   𝑥,𝑦

Proof of Theorem cvjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2614 . 2 (𝑥 = {𝑦𝑦𝑥} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥}))
2 df-clab 2607 . . 3 (𝑧 ∈ {𝑦𝑦𝑥} ↔ [𝑧 / 𝑦]𝑦𝑥)
3 elsb3 2432 . . 3 ([𝑧 / 𝑦]𝑦𝑥𝑧𝑥)
42, 3bitr2i 265 . 2 (𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥})
51, 4mpgbir 1724 1 𝑥 = {𝑦𝑦𝑥}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1481  [wsb 1878   ∈ wcel 1988  {cab 2606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613 This theorem is referenced by:  cnambfre  33429
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