Theorem cvjust 1464

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 952, which allows us to substitute a set variable for a class variable. See also cab 1456 and df-clab 1457. Note that this is not a rigorous justification, because cv 952 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 1463	. 2 |- (x = {y | y e. x} <-> A.z(z e. x <-> z e. {y | y e. x}))
2		df-clab 1457	. . 3 |- (z e. {y | y e. x} <-> [z / y]y e. x)
3		elsb3 1326	. . 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 985	1 |- x = {y | y e. x}

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  [wsbc 1166  {cab 1456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462

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