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

Definition df-cleq 2817
Description: Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

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-cleq 2817 an axiom.

See also comments under df-clab 2803, df-clel 2896, and abeq2 2948.

In the form of dfcleq 2818, this is called the "axiom of extensionality" 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 2803, df-cleq 2817, and df-clel 2896 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 2896. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)

Ref Expression
df-cleq.1 (𝑦 = 𝑧 ↔ ∀𝑢(𝑢𝑦𝑢𝑧))
df-cleq.2 (𝑡 = 𝑡 ↔ ∀𝑣(𝑣𝑡𝑣𝑡))
Ref Expression
df-cleq (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑡,𝑢,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑡,𝑢,𝑣

Detailed syntax breakdown of Definition df-cleq
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2wceq 1536 . 2 wff 𝐴 = 𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1535 . . . . 5 class 𝑥
65, 1wcel 2113 . . . 4 wff 𝑥𝐴
75, 2wcel 2113 . . . 4 wff 𝑥𝐵
86, 7wb 208 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4wal 1534 . 2 wff 𝑥(𝑥𝐴𝑥𝐵)
103, 9wb 208 1 wff (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  dfcleq  2818
  Copyright terms: Public domain W3C validator