Metamath Proof Explorer

Theorem predon

Description: The predecessor of an ordinal under _E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011)

Ref Expression
Assertion predon 
|- ( A e. On -> Pred ( _E , On , A ) = A )

Proof

Step Hyp Ref Expression
1 predep 
 |-  ( A e. On -> Pred ( _E , On , A ) = ( On i^i A ) )
2 onss 
 |-  ( A e. On -> A C_ On )
3 sseqin2 
 |-  ( A C_ On <-> ( On i^i A ) = A )
4 2 3 sylib 
 |-  ( A e. On -> ( On i^i A ) = A )
5 1 4 eqtrd 
 |-  ( A e. On -> Pred ( _E , On , A ) = A )