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Definition df-clel 2279
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 2276 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2276 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1954), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2270. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2904, clel3 2906, and clel4 2907.

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 1684 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1622 . . . . 5  class  x
65, 1wceq 1623 . . . 4  wff  x  =  A
75, 2wcel 1684 . . . 4  wff  x  e.  B
86, 7wa 358 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1528 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 176 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1  2343  eleq2  2344  clelab  2403  clabel  2404  nfel  2427  nfeld  2434  sbabel  2445  risset  2590  isset  2792  elex  2796  sbcabel  3068  ssel  3174  disjsn  3693  pwpw0  3763  pwsnALT  3822  mptpreima  5166  ballotlem2  23047  eldm3  24119
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