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Definition df-clel 2605
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 2602 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2602 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1984), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2596. Alternate definitions of 𝐴𝐵 (but that require either 𝐴 or 𝐵 to be a set) are shown by clel2 3308, clel3 3310, and clel4 3311.

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.

While the three class definitions df-clab 2596, df-cleq 2602, and df-clel 2605 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)

Assertion
Ref Expression
df-clel (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
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 1976 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1473 . . . . 5 class 𝑥
65, 1wceq 1474 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 1976 . . . 4 wff 𝑥𝐵
86, 7wa 382 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1694 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 194 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  eleq1d  2671  eleq2d  2672  eleq2dOLD  2673  eleq2dALT  2674  clelab  2734  clabel  2735  nfeld  2758  risset  3043  isset  3179  elex  3184  sbcabel  3482  ssel  3561  disjsn  4191  pwpw0  4283  pwsnALT  4361  mptpreima  5531  fi1uzind  13080  brfi1indALT  13083  fi1uzindOLD  13086  brfi1indALTOLD  13089  ballotlem2  29683  eldm3  30711  bj-clabel  31777  eliminable3a  31833  eliminable3b  31834  bj-eleq1w  31836  bj-eleq2w  31837  bj-denotes  31848  bj-issetwt  31849  bj-elissetv  31851  bj-ax8  31876  bj-df-clel  31877  bj-elsngl  31945
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