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Definition df-clel 2767
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 2764 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2764 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2153), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2758. Alternate definitions of 𝐴𝐵 (but that require either 𝐴 or 𝐵 to be a set) are shown by clel2 3490, clel3 3492, and clel4 3493.

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 2758, df-cleq 2764, and df-clel 2767 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 2145 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1630 . . . . 5 class 𝑥
65, 1wceq 1631 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 2145 . . . 4 wff 𝑥𝐵
86, 7wa 382 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1852 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 196 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  eleq1w  2833  eleq2w  2834  eleq1d  2835  eleq2d  2836  eleq2dALT  2837  clelab  2897  clabel  2898  nfeld  2922  risset  3210  isset  3359  elex  3364  sbcabel  3666  ssel  3746  disjsn  4384  pwpw0  4480  pwsnALT  4568  mptpreima  5771  fi1uzind  13480  brfi1indALT  13483  ballotlem2  30889  eldm3  31988  bj-clabel  33118  eliminable3a  33177  eliminable3b  33178  bj-denotes  33186  bj-issetwt  33187  bj-elissetv  33189  bj-ax8  33215  bj-df-clel  33216  bj-elsngl  33286
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