Theorem gchi 8905

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 7430	. . . . . . 7 |- Rel ~<
2	1	brrelexi 4990	. . . . . 6 |- ( B ~< ~P A -> B e. _V )
3	2	adantl 466	. . . . 5 |- ( ( A ~< B /\ B ~< ~P A ) -> B e. _V )
4		breq2 4407	. . . . . . 7 |- ( x = B -> ( A ~< x <-> A ~< B ) )
5		breq1 4406	. . . . . . 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 3164	. . . . 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 1588	. . . 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 8903	. . . . . 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 1186	1 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 965   A.wal 1368   = wceq 1370   E.wex 1587   e. wcel 1758   _Vcvv 3078   ~Pcpw 3971   class class class wbr 4403   ~< csdm 7422   Fincfn 7423  GCHcgch 8901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-dom 7425  df-sdom 7426  df-gch 8902

This theorem is referenced by:  gchen1  8906  gchen2  8907  gchpwdom  8951  gchaleph  8952