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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 1352, which allows us to substitute a setvar variable for a class variable. See also cab 2163 and df-clab 2164. Note that this is not a rigorous justification, because cv 1352 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 |
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cvjust |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2171 |
. 2
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2 | df-clab 2164 |
. . 3
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3 | elsb1 2155 |
. . 3
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4 | 2, 3 | bitr2i 185 |
. 2
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5 | 1, 4 | mpgbir 1453 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 |
This theorem is referenced by: (None) |
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