Theorem gchor 9449

Description: If A <_ B <_ ~P A, and A is an infinite GCH-set, then either A = B or B = ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

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

Proof of Theorem gchor

Step	Hyp	Ref	Expression
1		simprr 796	. . 3 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> B ~<_ ~P A )
2		brdom2 7985	. . 3 |- ( B ~<_ ~P A <-> ( B ~< ~P A \/ B ~~ ~P A ) )
3	1, 2	sylib 208	. 2 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A \/ B ~~ ~P A ) )
4		gchen1 9447	. . . . 5 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )
5	4	expr 643	. . . 4 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ A ~<_ B ) -> ( B ~< ~P A -> A ~~ B ) )
6	5	adantrr 753	. . 3 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A -> A ~~ B ) )
7	6	orim1d 884	. 2 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( ( B ~< ~P A \/ B ~~ ~P A ) -> ( A ~~ B \/ B ~~ ~P A ) ) )
8	3, 7	mpd 15	1 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( A ~~ B \/ B ~~ ~P A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   e. wcel 1990   ~Pcpw 4158   class class class wbr 4653   ~~ cen 7952   ~<_ cdom 7953   ~< 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-eu 2474  df-mo 2475  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-f1o 5895  df-en 7956  df-dom 7957  df-sdom 7958  df-gch 9443

This theorem is referenced by:  gchdomtri  9451  gchpwdom  9492