Description: Alternate definition of the class of hereditarily finite sets as the value of the cumulative hierarchy of sets function at _om . This definition is simpler but requires Infinity to work. (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 | r1funlim | |- ( Fun R1 /\ Lim dom R1 ) |
|
| 2 | 1 | simpli | |- Fun R1 |
| 3 | funiunfv | |- ( Fun R1 -> U_ x e. _om ( R1 ` x ) = U. ( R1 " _om ) ) |
|
| 4 | 2 3 | ax-mp | |- U_ x e. _om ( R1 ` x ) = U. ( R1 " _om ) |
| 5 | omex | |- _om e. _V |
|
| 6 | limom | |- Lim _om |
|
| 7 | r1lim | |- ( ( _om e. _V /\ Lim _om ) -> ( R1 ` _om ) = U_ x e. _om ( R1 ` x ) ) |
|
| 8 | 5 6 7 | mp2an | |- ( R1 ` _om ) = U_ x e. _om ( R1 ` x ) |
| 9 | df-hf | |- HF = U. ( R1 " _om ) |
|
| 10 | 4 8 9 | 3eqtr4ri | |- HF = ( R1 ` _om ) |