Theorem gchor

Description: If A <_ B <_ ~P A , and A is an infinite GCH-set, then either A = B or B = ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	gchor	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( A ~~ B \/ B ~~ ~P A ) )

Proof

Step	Hyp	Ref	Expression
1		simprr	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> B ~<_ ~P A )
2		brdom2	|- ( B ~<_ ~P A <-> ( B ~< ~P A \/ B ~~ ~P A ) )
3	1 2	sylib	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A \/ B ~~ ~P A ) )
4		gchen1	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )
5	4	expr	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ A ~<_ B ) -> ( B ~< ~P A -> A ~~ B ) )
6	5	adantrr	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A -> A ~~ B ) )
7	6	orim1d	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( ( B ~< ~P A \/ B ~~ ~P A ) -> ( A ~~ B \/ B ~~ ~P A ) ) )
8	3 7	mpd	|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( A ~~ B \/ B ~~ ~P A ) )