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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 1341, which allows us to substitute a setvar variable for a class variable. See also cab 2150 and df-clab 2151. Note that this is not a rigorous justification, because cv 1341 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 2158 | . 2 | |
2 | df-clab 2151 | . . 3 | |
3 | elsb3 2142 | . . 3 | |
4 | 2, 3 | bitr2i 184 | . 2 |
5 | 1, 4 | mpgbir 1440 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1342 wsb 1749 wcel 2135 cab 2150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 |
This theorem is referenced by: (None) |
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