Metamath Proof Explorer

Theorem 0sdom1dom

Description: Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004)

Ref Expression
Assertion 0sdom1dom 
|- ( (/) ~< A <-> 1o ~<_ A )

Proof

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 )