Metamath Proof Explorer

Theorem on0eln0

Description: An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004)

Ref Expression
Assertion on0eln0 
|- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )

Proof

Step Hyp Ref Expression
1 eloni 
 |-  ( A e. On -> Ord A )
2 ord0eln0 
 |-  ( Ord A -> ( (/) e. A <-> A =/= (/) ) )
3 1 2 syl 
 |-  ( A e. On -> ( (/) e. A <-> A =/= (/) ) )