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Definition df-clel 2434
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 2431 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2431 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2105), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2425. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 3074, clel3 3076, and clel4 3077.

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/mpeuni/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 1726 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1652 . . . . 5  class  x
65, 1wceq 1653 . . . 4  wff  x  =  A
75, 2wcel 1726 . . . 4  wff  x  e.  B
86, 7wa 360 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1551 . 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  2498  eleq2  2499  clelab  2558  clabel  2559  nfel  2582  nfeld  2589  sbabel  2600  risset  2755  isset  2962  elex  2966  sbcabel  3240  ssel  3344  disjsn  3870  pwpw0  3948  pwsnALT  4012  mptpreima  5366  brfi1uzind  11720  ballotlem2  24751  eldm3  25390
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