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Theorem cvjust 2821
 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 1657, which allows us to substitute a setvar variable for a class variable. See also cab 2812 and df-clab 2813. Note that this is not a rigorous justification, because cv 1657 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." See abid1 2950 for the version of cvjust 2821 extended to classes. (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 2820 . 2 (𝑥 = {𝑦𝑦𝑥} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥}))
2 df-clab 2813 . . 3 (𝑧 ∈ {𝑦𝑦𝑥} ↔ [𝑧 / 𝑦]𝑦𝑥)
3 elsb3 2569 . . 3 ([𝑧 / 𝑦]𝑦𝑥𝑧𝑥)
42, 3bitr2i 268 . 2 (𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥})
51, 4mpgbir 1900 1 𝑥 = {𝑦𝑦𝑥}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1658  [wsb 2069   ∈ wcel 2166  {cab 2812 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-12 2222  ax-13 2391  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819 This theorem is referenced by:  cnambfre  34002
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