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Definition df-clel 2252
Description: Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 2249 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2249 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2063), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2243. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2872, clel3 2874, and clel4 2875.

This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
df-clel  |-  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Detailed syntax breakdown of Definition df-clel
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
31, 2wcel 1621 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1618 . . . . 5  class  x
65, 1wceq 1619 . . . 4  wff  x  =  A
75, 2wcel 1621 . . . 4  wff  x  e.  B
86, 7wa 360 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1537 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 178 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1  2316  eleq2  2317  clelab  2376  clabel  2377  nfel  2400  nfeld  2407  sbabel  2418  risset  2563  isset  2761  elex  2765  sbcabel  3029  ssel  3135  disjsn  3653  pwpw0  3723  pwsnALT  3782  mptpreima  5139  ballotlem2  22994
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