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Definition df-clel 1465
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 1462 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 1462 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1323), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 1457.
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 955 . 2 wff A e. B
4 vx . . . . . 6 set x
54cv 952 . . . . 5 class x
65, 1wceq 953 . . . 4 wff x = A
75, 2wcel 955 . . . 4 wff x e. B
86, 7wa 223 . . 3 wff (x = A /\ x e. B)
98, 4wex 977 . 2 wff E.x(x = A /\ x e. B)
103, 9wb 146 1 wff (A e. B <-> E.x(x = A /\ x e. B))
Colors of variables: wff set class
This definition is referenced by:  eleq1 1526  eleq2 1527  hbel 1558  clelab 1573  clabel 1574  sbabel 1576  risset 1677  isset 1805  elisset 1808  sbcabel 1986  sbcel12g 2001  ssel 2053  pwpw0 2460  opelxp 3204  prnmadd 5072
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