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Metamath Proof Explorer

Theorem gchen1

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

Ref Expression
Assertion gchen1 
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )

Proof

Step Hyp Ref Expression
1 simprl 
 |-  ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~<_ B )
2 gchi 
 |-  ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )
3 2 3com23 
 |-  ( ( A e. GCH /\ B ~< ~P A /\ A ~< B ) -> A e. Fin )
4 3 3expia 
 |-  ( ( A e. GCH /\ B ~< ~P A ) -> ( A ~< B -> A e. Fin ) )
5 4 con3dimp 
 |-  ( ( ( A e. GCH /\ B ~< ~P A ) /\ -. A e. Fin ) -> -. A ~< B )
6 5 an32s 
 |-  ( ( ( A e. GCH /\ -. A e. Fin ) /\ B ~< ~P A ) -> -. A ~< B )
7 6 adantrl 
 |-  ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> -. A ~< B )
8 bren2 
 |-  ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )
9 1 7 8 sylanbrc 
 |-  ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )