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|Description: Define class abstraction
notation (so-called by Quine), also called a
"class builder" in the literature. and need not be distinct.
Definition 2.1 of [Quine] p. 16.
Typically, will have as a
free variable, and " " is read "the class of all sets
such that |
This is our first use of the
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 3167 which is used, for example, to convert elirrv 8044 to elirr 8045.
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 a "class term".
While the three class definitions df-clab 2443, df-cleq 2449, and df-clel 2452 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 http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 26-May-1993.)
|1||vx||. . . 4|
|2||1||cv 1394||. . 3|
|3||wph||. . . 4|
|4||vy||. . . 4|
|5||3, 4||cab 2442||. . 3|
|6||2, 5||wcel 1818||. 2|
|7||3, 4, 1||wsb 1739||. 2|
|8||6, 7||wb 184||1|
|Colors of variables: wff setvar class|
|This definition is referenced by: abid 2444 hbab1 2445 hbab 2447 cvjust 2451 abbiOLD 2589 cbvab 2598 cbvabOLD 2599 clelab 2601 clelabOLD 2602 nfabd2 2640 vjust 3110 dfsbcq2 3330 sbc8g 3335 unab 3764 inab 3765 difab 3766 csbab 3855 csbabgOLD 3856 exss 4715 iotaeq 5564 abrexex2g 6777 opabex3d 6778 opabex3 6779 abrexex2 6781 bj-hbab1 34356 bj-abbi 34361 bj-vjust 34372 eliminable1 34415 bj-vexwt 34430 bj-vexwvt 34432 bj-abfal 34474 bj-snsetex 34521|
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