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Metamath Proof Explorer

Theorem gchi

Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015)

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

Proof

Step Hyp Ref Expression
1 relsdom 
 |-  Rel ~<
2 1 brrelex1i 
 |-  ( B ~< ~P A -> B e. _V )
3 2 adantl 
 |-  ( ( A ~< B /\ B ~< ~P A ) -> B e. _V )
4 breq2 
 |-  ( x = B -> ( A ~< x <-> A ~< B ) )
5 breq1 
 |-  ( x = B -> ( x ~< ~P A <-> B ~< ~P A ) )
6 4 5 anbi12d 
 |-  ( x = B -> ( ( A ~< x /\ x ~< ~P A ) <-> ( A ~< B /\ B ~< ~P A ) ) )
7 6 spcegv 
 |-  ( B e. _V -> ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) ) )
8 3 7 mpcom 
 |-  ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) )
9 df-ex 
 |-  ( E. x ( A ~< x /\ x ~< ~P A ) <-> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
10 8 9 sylib 
 |-  ( ( A ~< B /\ B ~< ~P A ) -> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
11 elgch 
 |-  ( A e. GCH -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
12 11 ibi 
 |-  ( A e. GCH -> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) )
13 12 orcomd 
 |-  ( A e. GCH -> ( A. x -. ( A ~< x /\ x ~< ~P A ) \/ A e. Fin ) )
14 13 ord 
 |-  ( A e. GCH -> ( -. A. x -. ( A ~< x /\ x ~< ~P A ) -> A e. Fin ) )
15 10 14 syl5 
 |-  ( A e. GCH -> ( ( A ~< B /\ B ~< ~P A ) -> A e. Fin ) )
16 15 3impib 
 |-  ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )