Theorem cvjust 2731

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 1539, which allows to substitute a setvar variable for a class variable. See also cab 2714 and df-clab 2715. Note that this is not a rigorous justification, because cv 1539 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 2878 for the version of cvjust 2731 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2377. (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 2730	. 2 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥}))
2		df-clab 2715	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ [𝑧 / 𝑦]𝑦 ∈ 𝑥)
3		elsb1 2116	. . 3 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)
4	2, 3	bitr2i 276	. 2 ⊢ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥})
5	1, 4	mpgbir 1799	1 ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}

Syntax hints:   ↔ wb 206   = wceq 1540  [wsb 2064   ∈ wcel 2108  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729

This theorem is referenced by:  cnambfre  37675