Theorem cvjust 2348

Description: Every setvar 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 1641, which allows us to substitute a setvar variable for a class variable. See also cab 2339 and df-clab 2340. Note that this is not a rigorous justification, because cv 1641 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 2347	. 2 ⊢ (x = {y ∣ y ∈ x} ↔ ∀z(z ∈ x ↔ z ∈ {y ∣ y ∈ x}))
2		df-clab 2340	. . 3 ⊢ (z ∈ {y ∣ y ∈ x} ↔ [z / y]y ∈ x)
3		elsb1 2103	. . 3 ⊢ ([z / y]y ∈ x ↔ z ∈ x)
4	2, 3	bitr2i 241	. 2 ⊢ (z ∈ x ↔ z ∈ {y ∣ y ∈ x})
5	1, 4	mpgbir 1550	1 ⊢ x = {y ∣ y ∈ x}

Syntax hints:   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by: (None)