Description: Alternate definition of the class of hereditarily finite sets as the value of the cumulative hierarchy of sets function at _om . This characterization is simpler but requires the axiom of infinity to hold. (Contributed by BTernaryTau, 25-Jan-2026) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfhf2 | |- HF = ( R1 ` _om ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r1fun | |- Fun R1 |
|
| 2 | funiunfv | |- ( Fun R1 -> U_ x e. _om ( R1 ` x ) = U. ( R1 " _om ) ) |
|
| 3 | 1 2 | ax-mp | |- U_ x e. _om ( R1 ` x ) = U. ( R1 " _om ) |
| 4 | omex | |- _om e. _V |
|
| 5 | limom | |- Lim _om |
|
| 6 | r1lim | |- ( ( _om e. _V /\ Lim _om ) -> ( R1 ` _om ) = U_ x e. _om ( R1 ` x ) ) |
|
| 7 | 4 5 6 | mp2an | |- ( R1 ` _om ) = U_ x e. _om ( R1 ` x ) |
| 8 | df-hf | |- HF = U. ( R1 " _om ) |
|
| 9 | 3 7 8 | 3eqtr4ri | |- HF = ( R1 ` _om ) |