MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-clel Structured version   Visualization version   GIF version

Definition df-clel 2892
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.

The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-clel 2892 an axiom.

See also comments under df-clab 2800, df-cleq 2814, and abeq2 2944.

Alternate characterizations of 𝐴𝐵 when either 𝐴 or 𝐵 is a set are given by clel2 3630, clel3 3632, and clel4 3633.

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 2800, df-cleq 2814, and df-clel 2892 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 2892. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.)

Hypotheses
Ref Expression
df-clel.1 (𝑦𝑧 ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧))
df-clel.2 (𝑡𝑡 ↔ ∃𝑣(𝑣 = 𝑡𝑣𝑡))
Assertion
Ref Expression
df-clel (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑡,𝑢,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑡,𝑢,𝑣

Detailed syntax breakdown of Definition df-clel
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2wcel 2115 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1537 . . . . 5 class 𝑥
65, 1wceq 1538 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 2115 . . . 4 wff 𝑥𝐵
86, 7wa 399 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1781 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 209 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  dfclel  2893
  Copyright terms: Public domain W3C validator