Description: Alternate proof of 0sdom1dom , shorter but requiring ax-un . (Contributed by NM, 28-Sep-2004) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 0sdom1domALT | |- ( (/) ~< A <-> 1o ~<_ A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | peano1 | |- (/) e. _om |
|
| 2 | sucdom | |- ( (/) e. _om -> ( (/) ~< A <-> suc (/) ~<_ A ) ) |
|
| 3 | 1 2 | ax-mp | |- ( (/) ~< A <-> suc (/) ~<_ A ) |
| 4 | df-1o | |- 1o = suc (/) |
|
| 5 | 4 | breq1i | |- ( 1o ~<_ A <-> suc (/) ~<_ A ) |
| 6 | 3 5 | bitr4i | |- ( (/) ~< A <-> 1o ~<_ A ) |