Theorem cvjust 2726

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 1540, which allows to substitute a setvar variable for a class variable. See also cab 2709 and df-clab 2710. Note that this is not a rigorous justification, because cv 1540 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 2870 for the version of cvjust 2726 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2371. (Revised by Wolf Lammen, 4-May-2023.)

Assertion

Ref	Expression
cvjust	⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}

Distinct variable group:   𝑥,𝑦

Proof of Theorem cvjust

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2725	. 2 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥}))
2		df-clab 2710	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ [𝑧 / 𝑦]𝑦 ∈ 𝑥)
3		elsb1 2114	. . 3 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)
4	2, 3	bitr2i 275	. 2 ⊢ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥})
5	1, 4	mpgbir 1801	1 ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}

Syntax hints:   ↔ wb 205   = wceq 1541  [wsb 2067   ∈ wcel 2106  {cab 2709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724

This theorem is referenced by:  cnambfre  36524