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Definition df-clel 2249
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 2246 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2246 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2060), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2240. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2841, clel3 2843, and clel4 2844.

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  2313  eleq2  2314  clelab  2369  clabel  2370  nfel  2393  nfeld  2400  sbabel  2411  risset  2552  isset  2731  elex  2735  sbcabel  2998  ssel  3097  disjsn  3597  pwpw0  3663  pwsnALT  3722  mptpreima  5072
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