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Theorem cvjust 2766
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 1630, which allows us to substitute a setvar variable for a class variable. See also cab 2757 and df-clab 2758. Note that this is not a rigorous justification, because cv 1630 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 2765 . 2 (𝑥 = {𝑦𝑦𝑥} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥}))
2 df-clab 2758 . . 3 (𝑧 ∈ {𝑦𝑦𝑥} ↔ [𝑧 / 𝑦]𝑦𝑥)
3 elsb3 2583 . . 3 ([𝑧 / 𝑦]𝑦𝑥𝑧𝑥)
42, 3bitr2i 265 . 2 (𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥})
51, 4mpgbir 1874 1 𝑥 = {𝑦𝑦𝑥}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  [wsb 2049  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764
This theorem is referenced by:  cnambfre  33790
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