Theorem cvjust 1879

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 1297, which allows us to substitute a set variable for a class variable. See also cab 1871 and df-clab 1872. Note that this is not a rigorous justification, because cv 1297 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 1878	. 2 |- (x = {y | y e. x} <-> A.z(z e. x <-> z e. {y | y e. x}))
2		df-clab 1872	. . 3 |- (z e. {y | y e. x} <-> [z / y]y e. x)
3		elsb3 1718	. . 3 |- ([z / y]y e. x <-> z e. x)
4	2, 3	bitr2i 191	. 2 |- (z e. x <-> z e. {y | y e. x})
5	1, 4	mpgbir 1334	1 |- x = {y | y e. x}

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877

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