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Metamath Proof Explorer

Theorem cvjust

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 , which allows to substitute a setvar variable for a class variable. See also cab and df-clab . Note that this is not a rigorous justification, because cv 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 for the version of cvjust extended to classes. (Contributed by NM, 7-Nov-2006) Avoid ax-13 . (Revised by Wolf Lammen, 4-May-2023)

Ref Expression
Assertion cvjust ⊢ x = y | y ∈ x

Proof

Step Hyp Ref Expression
1 dfcleq ⊢ x = y | y ∈ x ↔ ∀ z z ∈ x ↔ z ∈ y | y ∈ x
2 df-clab ⊢ z ∈ y | y ∈ x ↔ z y y ∈ x
3 elsb1 ⊢ z y y ∈ x ↔ z ∈ x
4 2 3 bitr2i ⊢ z ∈ x ↔ z ∈ y | y ∈ x
5 1 4 mpgbir ⊢ x = y | y ∈ x