Theorem orddif

Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998)

	Ref	Expression
Assertion	orddif	|- ( Ord A -> A = ( suc A \ { A } ) )

Proof

Step	Hyp	Ref	Expression
1		orddisj	|- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3	|- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc	|- suc A = ( A u. { A } )
4	3	difeq1i	|- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2	|- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4 5	eqtri	|- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i	|- ( A = ( suc A \ { A } ) <-> A = ( A \ { A } ) )
8	2 7	bitr4i	|- ( ( A i^i { A } ) = (/) <-> A = ( suc A \ { A } ) )
9	1 8	sylib	|- ( Ord A -> A = ( suc A \ { A } ) )