Theorem gchi 9005

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 7525	. . . . . . 7 |- Rel ~<
2	1	brrelexi 5030	. . . . . 6 |- ( B ~< ~P A -> B e. _V )
3	2	adantl 466	. . . . 5 |- ( ( A ~< B /\ B ~< ~P A ) -> B e. _V )
4		breq2 4441	. . . . . . 7 |- ( x = B -> ( A ~< x <-> A ~< B ) )
5		breq1 4440	. . . . . . 7 |- ( x = B -> ( x ~< ~P A <-> B ~< ~P A ) )
6	4, 5	anbi12d 710	. . . . . 6 |- ( x = B -> ( ( A ~< x /\ x ~< ~P A ) <-> ( A ~< B /\ B ~< ~P A ) ) )
7	6	spcegv 3181	. . . . 5 |- ( B e. _V -> ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) ) )
8	3, 7	mpcom 36	. . . 4 |- ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) )
9		df-ex 1600	. . . 4 |- ( E. x ( A ~< x /\ x ~< ~P A ) <-> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
10	8, 9	sylib 196	. . 3 |- ( ( A ~< B /\ B ~< ~P A ) -> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
11		elgch 9003	. . . . . 6 |- ( A e. GCH -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
12	11	ibi 241	. . . . 5 |- ( A e. GCH -> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) )
13	12	orcomd 388	. . . 4 |- ( A e. GCH -> ( A. x -. ( A ~< x /\ x ~< ~P A ) \/ A e. Fin ) )
14	13	ord 377	. . 3 |- ( A e. GCH -> ( -. A. x -. ( A ~< x /\ x ~< ~P A ) -> A e. Fin ) )
15	10, 14	syl5 32	. 2 |- ( A e. GCH -> ( ( A ~< B /\ B ~< ~P A ) -> A e. Fin ) )
16	15	3impib 1195	1 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 974   A.wal 1381   = wceq 1383   E.wex 1599   e. wcel 1804   _Vcvv 3095   ~Pcpw 3997   class class class wbr 4437   ~< csdm 7517   Fincfn 7518  GCHcgch 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-dom 7520  df-sdom 7521  df-gch 9002

This theorem is referenced by:  gchen1  9006  gchen2  9007  gchpwdom  9051  gchaleph  9052