MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-clel Unicode version

Definition df-clel 2292
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 2289 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2289 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1967), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2283. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2917, clel3 2919, and clel4 2920.

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 1696 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1631 . . . . 5  class  x
65, 1wceq 1632 . . . 4  wff  x  =  A
75, 2wcel 1696 . . . 4  wff  x  e.  B
86, 7wa 358 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1531 . 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  2356  eleq2  2357  clelab  2416  clabel  2417  nfel  2440  nfeld  2447  sbabel  2458  risset  2603  isset  2805  elex  2809  sbcabel  3081  ssel  3187  disjsn  3706  pwpw0  3779  pwsnALT  3838  mptpreima  5182  ballotlem2  23063  eldm3  24190
  Copyright terms: Public domain W3C validator