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Description: Two ways to say
" is a set":
A class is a member
of the
universal class (see df-v 2688) if and only if the class
exists (i.e. there exists some set equal to class ).
Theorem 6.9 of [Quine] p. 43.
Notational convention: We will use the
notational device " " to mean
" is a set"
very
frequently, for example in uniex 4359. Note the when is not a set,
it is called a proper class. In some theorems, such as uniexg 4361, in
order to shorten certain proofs we use the more general antecedent
instead of to
mean " is a
set."
Note that a constant is implicitly considered distinct from all variables. This is why is not included in the distinct variable list, even though df-clel 2135 requires that the expression substituted for not contain . (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.) |
Ref | Expression |
---|---|
isset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2135 | . 2 | |
2 | vex 2689 | . . . 4 | |
3 | 2 | biantru 300 | . . 3 |
4 | 3 | exbii 1584 | . 2 |
5 | 1, 4 | bitr4i 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2686 |
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-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 |
This theorem is referenced by: issetf 2693 isseti 2694 issetri 2695 elex 2697 elisset 2700 vtoclg1f 2745 ceqex 2812 eueq 2855 moeq 2859 mosubt 2861 ru 2908 sbc5 2932 snprc 3588 vprc 4060 opelopabsb 4182 eusvnfb 4375 euiotaex 5104 fvmptdf 5508 fvmptdv2 5510 fmptco 5586 brabvv 5817 ovmpodf 5902 ovi3 5907 tfrlemibxssdm 6224 tfr1onlembxssdm 6240 tfrcllembxssdm 6253 ecexr 6434 snexxph 6838 bj-vprc 13094 bj-vnex 13096 bj-2inf 13136 |
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