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Definition df-clel 2510
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 2507 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2507 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1946), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2501. Alternate definitions of 𝐴𝐵 (but that require either 𝐴 or 𝐵 to be a set) are shown by clel2 3213, clel3 3215, and clel4 3216.

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 2501, df-cleq 2507, and df-clel 2510 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 1938 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1473 . . . . 5 class 𝑥
65, 1wceq 1474 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 1938 . . . 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  2576  eleq2d  2577  eleq2dOLD  2578  eleq2dALT  2579  clelab  2639  clabel  2640  nfeld  2663  risset  2948  isset  3084  elex  3089  sbcabel  3387  ssel  3466  disjsn  4095  pwpw0  4187  pwsnALT  4265  mptpreima  5435  fi1uzind  12991  brfi1indALT  12994  fi1uzindOLD  12997  brfi1indALTOLD  13000  ballotlem2  29685  eldm3  30748  bj-clabel  31813  eliminable3a  31869  eliminable3b  31870  bj-eleq1w  31872  bj-eleq2w  31873  bj-denotes  31884  bj-issetwt  31885  bj-elissetv  31887  bj-ax8  31912  bj-df-clel  31913  bj-elsngl  31981
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