Metamath Proof Explorer


Theorem onnbtwn

Description: There is no set between an ordinal number and its successor. Proposition 7.25 of TakeutiZaring p. 41. (Contributed by NM, 9-Jun-1994)

Ref Expression
Assertion onnbtwn
|- ( A e. On -> -. ( A e. B /\ B e. suc A ) )

Proof

Step Hyp Ref Expression
1 eloni
 |-  ( A e. On -> Ord A )
2 ordnbtwn
 |-  ( Ord A -> -. ( A e. B /\ B e. suc A ) )
3 1 2 syl
 |-  ( A e. On -> -. ( A e. B /\ B e. suc A ) )