Theorem cvjust 1474

Description: Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 957, which allows us to substitute a set variable for a class variable. See also cab 1466 and df-clab 1467. Note that this is not a rigorous justification, because cv 957 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."

Assertion

Ref	Expression
cvjust	|- x = {y | y e. x}

Distinct variable group:   x,y

Proof of Theorem cvjust

Step	Hyp	Ref	Expression
1		dfcleq 1473	. 2 |- (x = {y | y e. x} <-> A.z(z e. x <-> z e. {y | y e. x}))
2		df-clab 1467	. . 3 |- (z e. {y | y e. x} <-> [z / y]y e. x)
3		elsb3 1333	. . 3 |- ([z / y]y e. x <-> z e. x)
4	2, 3	bitr2 174	. 2 |- (z e. x <-> z e. {y | y e. x})
5	1, 4	mpgbir 990	1 |- x = {y | y e. x}

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  [wsbc 1172  {cab 1466

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472

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