Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2817 an axiom. See also comments under df-clab 2716, df-cleq 2730, and abeq2 2870. Alternate characterizations of 𝐴 ∈ 𝐵 when either 𝐴 or 𝐵 is a set are given by clel2g 3579, clel3g 3582, and clel4g 3584. 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 2716, df-cleq 2730, and df-clel 2817 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 2817. (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 2111 | . 2 wff 𝐴 ∈ 𝐵 |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1542 | . . . . 5 class 𝑥 |
6 | 5, 1 | wceq 1543 | . . . 4 wff 𝑥 = 𝐴 |
7 | 5, 2 | wcel 2111 | . . . 4 wff 𝑥 ∈ 𝐵 |
8 | 6, 7 | wa 399 | . . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) |
9 | 8, 4 | wex 1787 | . 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 2818 |
Copyright terms: Public domain | W3C validator |