Description: The sets whose transitive closures are sets are precisely the well-founded sets, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ttcwf3 | |- ( TC+ A e. _V <-> A e. U. ( R1 " On ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ttcwf2 | |- ( TC+ A e. _V <-> TC+ A e. U. ( R1 " On ) ) |
|
| 2 | ttcwf | |- ( A e. U. ( R1 " On ) <-> TC+ A e. U. ( R1 " On ) ) |
|
| 3 | 1 2 | bitr4i | |- ( TC+ A e. _V <-> A e. U. ( R1 " On ) ) |