Definition df-clab 2492

Description: Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. 𝑥 and 𝑦 need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, 𝜑 will have 𝑦 as a free variable, and "{𝑦 ∣ 𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true." We do not define {𝑦 ∣ 𝜑} in isolation but only as part of an expression that extends or "overloads" the ∈ relationship.

This is our first use of the ∈ symbol to connect classes instead of sets. The syntax definition wcel 1937, which extends or "overloads" the wel 1938 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2498 and df-clel 2501, we introduce a new kind of variable (class variable) that can be substituted with expressions such as {𝑦 ∣ 𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1467 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2500 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2614 for a quick overview).

Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is done with theorems such as vtoclg 3128 which is used, for example, to convert elirrv 8197 to elirr 8198.

This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction {𝑦 ∣ 𝜑} a "class term".

While the three class definitions df-clab 2492, df-cleq 2498, and df-clel 2501 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-clab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2	1	cv 1467	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2491	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 1937	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1828	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 191	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2493  hbab1  2494  hbab  2496  cvjust  2500  cbvab  2628  clelab  2630  nfabd2  2665  vjust  3067  dfsbcq2  3294  sbc8g  3299  unab  3736  inab  3737  difab  3738  abtru  3785  csbab  3837  exss  4704  iotaeq  5605  abrexex2g  6847  opabex3d  6848  opabex3  6849  abrexex2  6851  bj-hbab1  31568  bj-abbi  31572  bj-vjust  31583  eliminable1  31632  bj-vexwt  31647  bj-vexwvt  31649  bj-abfal  31693  bj-snsetex  31743  bj-vjust2  31805  csbabgOLD  37559