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Theorem cvjust 2500
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 1467, which allows us to substitute a setvar variable for a class variable. See also cab 2491 and df-clab 2492. Note that this is not a rigorous justification, because cv 1467 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 2499 . 2 (𝑥 = {𝑦𝑦𝑥} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥}))
2 df-clab 2492 . . 3 (𝑧 ∈ {𝑦𝑦𝑥} ↔ [𝑧 / 𝑦]𝑦𝑥)
3 elsb3 2317 . . 3 ([𝑧 / 𝑦]𝑦𝑥𝑧𝑥)
42, 3bitr2i 260 . 2 (𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥})
51, 4mpgbir 1702 1 𝑥 = {𝑦𝑦𝑥}
Colors of variables: wff setvar class
Syntax hints:  wb 191   = wceq 1468  [wsb 1828  wcel 1937  {cab 2491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498
This theorem is referenced by:  cnambfre  32227
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