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Theorem cvjust 2727
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 1541, which allows to substitute a setvar variable for a class variable. See also cab 2710 and df-clab 2711. Note that this is not a rigorous justification, because cv 1541 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 2871 for the version of cvjust 2727 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2372. (Revised by Wolf Lammen, 4-May-2023.)
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 2726 . 2 (𝑥 = {𝑦𝑦𝑥} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥}))
2 df-clab 2711 . . 3 (𝑧 ∈ {𝑦𝑦𝑥} ↔ [𝑧 / 𝑦]𝑦𝑥)
3 elsb1 2115 . . 3 ([𝑧 / 𝑦]𝑦𝑥𝑧𝑥)
42, 3bitr2i 276 . 2 (𝑧𝑥𝑧 ∈ {𝑦𝑦𝑥})
51, 4mpgbir 1802 1 𝑥 = {𝑦𝑦𝑥}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  [wsb 2068  wcel 2107  {cab 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725
This theorem is referenced by:  cnambfre  36536
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