Theorem gchi 9446

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.)

Assertion

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

Proof of Theorem gchi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 7962	. . . . . . 7 |- Rel ~<
2	1	brrelexi 5158	. . . . . 6 |- ( B ~< ~P A -> B e. _V )
3	2	adantl 482	. . . . 5 |- ( ( A ~< B /\ B ~< ~P A ) -> B e. _V )
4		breq2 4657	. . . . . . 7 |- ( x = B -> ( A ~< x <-> A ~< B ) )
5		breq1 4656	. . . . . . 7 |- ( x = B -> ( x ~< ~P A <-> B ~< ~P A ) )
6	4, 5	anbi12d 747	. . . . . 6 |- ( x = B -> ( ( A ~< x /\ x ~< ~P A ) <-> ( A ~< B /\ B ~< ~P A ) ) )
7	6	spcegv 3294	. . . . 5 |- ( B e. _V -> ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) ) )
8	3, 7	mpcom 38	. . . 4 |- ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) )
9		df-ex 1705	. . . 4 |- ( E. x ( A ~< x /\ x ~< ~P A ) <-> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
10	8, 9	sylib 208	. . 3 |- ( ( A ~< B /\ B ~< ~P A ) -> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
11		elgch 9444	. . . . . 6 |- ( A e. GCH -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
12	11	ibi 256	. . . . 5 |- ( A e. GCH -> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) )
13	12	orcomd 403	. . . 4 |- ( A e. GCH -> ( A. x -. ( A ~< x /\ x ~< ~P A ) \/ A e. Fin ) )
14	13	ord 392	. . 3 |- ( A e. GCH -> ( -. A. x -. ( A ~< x /\ x ~< ~P A ) -> A e. Fin ) )
15	10, 14	syl5 34	. 2 |- ( A e. GCH -> ( ( A ~< B /\ B ~< ~P A ) -> A e. Fin ) )
16	15	3impib 1262	1 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   ~< csdm 7954   Fincfn 7955  GCHcgch 9442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dom 7957  df-sdom 7958  df-gch 9443

This theorem is referenced by:  gchen1  9447  gchen2  9448  gchpwdom  9492  gchaleph  9493