Theorem cvjust 2191

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 1363, which allows us to substitute a setvar variable for a class variable. See also cab 2182 and df-clab 2183. Note that this is not a rigorous justification, because cv 1363 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 = {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 2190	. 2 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥}))
2		df-clab 2183	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥} ↔ [𝑧 / 𝑦]𝑦 ∈ 𝑥)
3		elsb1 2174	. . 3 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)
4	2, 3	bitr2i 185	. 2 ⊢ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 ∈ 𝑥})
5	1, 4	mpgbir 1467	1 ⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}

Syntax hints:   ↔ wb 105   = wceq 1364  [wsb 1776   ∈ wcel 2167  {cab 2182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189

This theorem is referenced by: (None)