Theorem gchi 8455

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 7075	. . . . . . 7 |- Rel ~<
2	1	brrelexi 4877	. . . . . 6 |- ( B ~< ~P A -> B e. _V )
3	2	adantl 453	. . . . 5 |- ( ( A ~< B /\ B ~< ~P A ) -> B e. _V )
4		breq2 4176	. . . . . . 7 |- ( x = B -> ( A ~< x <-> A ~< B ) )
5		breq1 4175	. . . . . . 7 |- ( x = B -> ( x ~< ~P A <-> B ~< ~P A ) )
6	4, 5	anbi12d 692	. . . . . 6 |- ( x = B -> ( ( A ~< x /\ x ~< ~P A ) <-> ( A ~< B /\ B ~< ~P A ) ) )
7	6	spcegv 2997	. . . . 5 |- ( B e. _V -> ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) ) )
8	3, 7	mpcom 34	. . . 4 |- ( ( A ~< B /\ B ~< ~P A ) -> E. x ( A ~< x /\ x ~< ~P A ) )
9		df-ex 1548	. . . 4 |- ( E. x ( A ~< x /\ x ~< ~P A ) <-> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
10	8, 9	sylib 189	. . 3 |- ( ( A ~< B /\ B ~< ~P A ) -> -. A. x -. ( A ~< x /\ x ~< ~P A ) )
11		elgch 8453	. . . . . 6 |- ( A e. GCH -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
12	11	ibi 233	. . . . 5 |- ( A e. GCH -> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) )
13	12	orcomd 378	. . . 4 |- ( A e. GCH -> ( A. x -. ( A ~< x /\ x ~< ~P A ) \/ A e. Fin ) )
14	13	ord 367	. . 3 |- ( A e. GCH -> ( -. A. x -. ( A ~< x /\ x ~< ~P A ) -> A e. Fin ) )
15	10, 14	syl5 30	. 2 |- ( A e. GCH -> ( ( A ~< B /\ B ~< ~P A ) -> A e. Fin ) )
16	15	3impib 1151	1 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 358   /\ wa 359   /\ w3a 936   A.wal 1546   E.wex 1547   = wceq 1649   e. wcel 1721   _Vcvv 2916   ~Pcpw 3759   class class class wbr 4172   ~< csdm 7067   Fincfn 7068  GCHcgch 8451

This theorem is referenced by:  gchen1  8456  gchen2  8457  gchpwdom  8505  gchaleph  8506

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-dom 7070  df-sdom 7071  df-gch 8452