Definition df-clel 2137

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 2134 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2134 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1979), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2129. Alternate definitions of A e. B (but that require either A or B to be a set) are shown by clel2 2636, clel3 2638, and clel4 2639.

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

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	wcel 1588	. 2 wff A e. B
4		vx	. . . . . 6 set x
5	4	cv 1585	. . . . 5 class x
6	5, 1	wceq 1586	. . . 4 wff x = A
7	5, 2	wcel 1588	. . . 4 wff x e. B
8	6, 7	wa 337	. . 3 wff (x = A /\ x e. B)
9	8, 4	wex 1615	. 2 wff E.x(x = A /\ x e. B)
10	3, 9	wb 219	1 wff (A e. B <-> E.x(x = A /\ x e. B))

This definition is referenced by:  eleq1 2204  eleq2 2205  hbel 2246  clelab 2262  clabel 2263  sbabel 2265  risset 2395  isset 2542  elisset 2547  sbcabel 2766  ssel 2846  disjsn 3280  pwpw0 3326  pwsnALT 3362  prnmadd 6618  prtlem16 17096

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