Theorem domfin4

Description: A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015)

	Ref	Expression
Assertion	domfin4	|- ( ( A e. Fin4 /\ B ~<_ A ) -> B e. Fin4 )

Proof

Step	Hyp	Ref	Expression
1		domeng	|- ( A e. Fin4 -> ( B ~<_ A <-> E. x ( B ~~ x /\ x C_ A ) ) )
2	1	biimpa	|- ( ( A e. Fin4 /\ B ~<_ A ) -> E. x ( B ~~ x /\ x C_ A ) )
3		ensym	|- ( B ~~ x -> x ~~ B )
4	3	ad2antrl	|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> x ~~ B )
5		ssfin4	|- ( ( A e. Fin4 /\ x C_ A ) -> x e. Fin4 )
6	5	ad2ant2rl	|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> x e. Fin4 )
7		fin4en1	|- ( x ~~ B -> ( x e. Fin4 -> B e. Fin4 ) )
8	4 6 7	sylc	|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> B e. Fin4 )
9	2 8	exlimddv	|- ( ( A e. Fin4 /\ B ~<_ A ) -> B e. Fin4 )