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Mirrors > Home > MPE Home > Th. List > df-clel | Structured version Visualization version GIF version |
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 𝐴 ∈ 𝐵 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.) |
Ref | Expression |
---|---|
df-clel.1 | ⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) |
df-clel.2 | ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) |
Ref | Expression |
---|---|
df-clel | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | 1, 2 | wcel 2115 | . 2 wff 𝐴 ∈ 𝐵 |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1537 | . . . . 5 class 𝑥 |
6 | 5, 1 | wceq 1538 | . . . 4 wff 𝑥 = 𝐴 |
7 | 5, 2 | wcel 2115 | . . . 4 wff 𝑥 ∈ 𝐵 |
8 | 6, 7 | wa 399 | . . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) |
9 | 8, 4 | wex 1781 | . 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) |
10 | 3, 9 | wb 209 | 1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfclel 2893 |
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