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Mirrors > Home > ILE Home > Th. List > cvjust | Unicode version |
Description: Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1283, which allows us to substitute a setvar variable for a class variable. See also cab 2067 and df-clab 2068. Note that this is not a rigorous justification, because cv 1283 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.) |
Ref | Expression |
---|---|
cvjust |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2075 | . 2 | |
2 | df-clab 2068 | . . 3 | |
3 | elsb3 1893 | . . 3 | |
4 | 2, 3 | bitr2i 183 | . 2 |
5 | 1, 4 | mpgbir 1382 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wceq 1284 wcel 1433 wsb 1685 cab 2067 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 |
This theorem is referenced by: (None) |
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