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Metamath Proof Explorer

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 ) )