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Mirrors > Home > NFE Home > Th. List > cvjust | Unicode version |
Description: Every setvar 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 1641, which allows us to substitute a setvar variable for a class variable. See also cab 2339 and df-clab 2340. Note that this is not a rigorous justification, because cv 1641 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 2347 | . 2 | |
2 | df-clab 2340 | . . 3 | |
3 | elsb3 2103 | . . 3 | |
4 | 2, 3 | bitr2i 241 | . 2 |
5 | 1, 4 | mpgbir 1550 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wceq 1642 wsb 1648 wcel 1710 cab 2339 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 |
This theorem is referenced by: (None) |
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