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Definition df-clel 2744
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 2741 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2741 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2135), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2735. Alternate definitions of 𝐴𝐵 (but that require either 𝐴 or 𝐵 to be a set) are shown by clel2 3467, clel3 3469, and clel4 3470.

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 2735, df-cleq 2741, and df-clel 2744 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 2127 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1619 . . . . 5 class 𝑥
65, 1wceq 1620 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 2127 . . . 4 wff 𝑥𝐵
86, 7wa 383 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1841 . 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  2810  eleq2w  2811  eleq1d  2812  eleq2d  2813  eleq2dALT  2814  clelab  2874  clabel  2875  nfeld  2899  risset  3188  isset  3335  elex  3340  sbcabel  3646  ssel  3726  disjsn  4378  pwpw0  4477  pwsnALT  4569  mptpreima  5777  fi1uzind  13442  brfi1indALT  13445  ballotlem2  30830  eldm3  31929  bj-clabel  33060  eliminable3a  33121  eliminable3b  33122  bj-denotes  33135  bj-issetwt  33136  bj-elissetv  33138  bj-ax8  33164  bj-df-clel  33165  bj-elsngl  33233
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