Theorem phplem1

Description: Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998)

	Ref	Expression
Assertion	phplem1	|- ( ( A e. _om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )

Proof

Step	Hyp	Ref	Expression
1		nnord	|- ( A e. _om -> Ord A )
2		nordeq	|- ( ( Ord A /\ B e. A ) -> A =/= B )
3		disjsn2	|- ( A =/= B -> ( { A } i^i { B } ) = (/) )
4	2 3	syl	|- ( ( Ord A /\ B e. A ) -> ( { A } i^i { B } ) = (/) )
5	1 4	sylan	|- ( ( A e. _om /\ B e. A ) -> ( { A } i^i { B } ) = (/) )
6		undif4	|- ( ( { A } i^i { B } ) = (/) -> ( { A } u. ( A \ { B } ) ) = ( ( { A } u. A ) \ { B } ) )
7		df-suc	|- suc A = ( A u. { A } )
8	7	equncomi	|- suc A = ( { A } u. A )
9	8	difeq1i	|- ( suc A \ { B } ) = ( ( { A } u. A ) \ { B } )
10	6 9	eqtr4di	|- ( ( { A } i^i { B } ) = (/) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
11	5 10	syl	|- ( ( A e. _om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )