Description: A family of subgroups indexed by a proper class cannot be a family of subgroups for an internal direct product. (Contributed by AV, 13-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | dprddomprc | |- ( dom S e/ _V -> -. G dom DProd S ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel | |- ( dom S e/ _V <-> -. dom S e. _V ) |
|
2 | dmexg | |- ( S e. _V -> dom S e. _V ) |
|
3 | 2 | con3i | |- ( -. dom S e. _V -> -. S e. _V ) |
4 | 1 3 | sylbi | |- ( dom S e/ _V -> -. S e. _V ) |
5 | reldmdprd | |- Rel dom DProd |
|
6 | 5 | brrelex2i | |- ( G dom DProd S -> S e. _V ) |
7 | 4 6 | nsyl | |- ( dom S e/ _V -> -. G dom DProd S ) |