Theorem gchdomtri 9451

Description: Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9503. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gchdomtri	|- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A ~<_ B \/ B ~<_ A ) )

Proof of Theorem gchdomtri

Step	Hyp	Ref	Expression
1		sdomdom 7983	. . . . 5 |- ( A ~< B -> A ~<_ B )
2	1	con3i 150	. . . 4 |- ( -. A ~<_ B -> -. A ~< B )
3		reldom 7961	. . . . . . 7 |- Rel ~<_
4	3	brrelexi 5158	. . . . . 6 |- ( B ~<_ ~P A -> B e. _V )
5	4	3ad2ant3 1084	. . . . 5 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> B e. _V )
6		fidomtri2 8820	. . . . 5 |- ( ( B e. _V /\ A e. Fin ) -> ( B ~<_ A <-> -. A ~< B ) )
7	5, 6	sylan 488	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A e. Fin ) -> ( B ~<_ A <-> -. A ~< B ) )
8	2, 7	syl5ibr 236	. . 3 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A e. Fin ) -> ( -. A ~<_ B -> B ~<_ A ) )
9	8	orrd 393	. 2 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
10		simp1 1061	. . . . 5 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> A e. GCH )
11	10	adantr 481	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> A e. GCH )
12		simpr 477	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> -. A e. Fin )
13		cdadom3 9010	. . . . . 6 |- ( ( A e. GCH /\ B e. _V ) -> A ~<_ ( A +c B ) )
14	10, 5, 13	syl2anc 693	. . . . 5 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> A ~<_ ( A +c B ) )
15	14	adantr 481	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> A ~<_ ( A +c B ) )
16		cdalepw 9018	. . . . . 6 |- ( ( ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )
17	16	3adant1 1079	. . . . 5 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A +c B ) ~<_ ~P A )
18	17	adantr 481	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> ( A +c B ) ~<_ ~P A )
19		gchor 9449	. . . 4 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ ( A +c B ) /\ ( A +c B ) ~<_ ~P A ) ) -> ( A ~~ ( A +c B ) \/ ( A +c B ) ~~ ~P A ) )
20	11, 12, 15, 18, 19	syl22anc 1327	. . 3 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> ( A ~~ ( A +c B ) \/ ( A +c B ) ~~ ~P A ) )
21		cdadom3 9010	. . . . . . . . 9 |- ( ( B e. _V /\ A e. GCH ) -> B ~<_ ( B +c A ) )
22	5, 10, 21	syl2anc 693	. . . . . . . 8 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> B ~<_ ( B +c A ) )
23		cdacomen 9003	. . . . . . . 8 |- ( B +c A ) ~~ ( A +c B )
24		domentr 8015	. . . . . . . 8 |- ( ( B ~<_ ( B +c A ) /\ ( B +c A ) ~~ ( A +c B ) ) -> B ~<_ ( A +c B ) )
25	22, 23, 24	sylancl 694	. . . . . . 7 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> B ~<_ ( A +c B ) )
26		domen2 8103	. . . . . . 7 |- ( A ~~ ( A +c B ) -> ( B ~<_ A <-> B ~<_ ( A +c B ) ) )
27	25, 26	syl5ibrcom 237	. . . . . 6 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A ~~ ( A +c B ) -> B ~<_ A ) )
28	27	imp 445	. . . . 5 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A ~~ ( A +c B ) ) -> B ~<_ A )
29	28	olcd 408	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ A ~~ ( A +c B ) ) -> ( A ~<_ B \/ B ~<_ A ) )
30		simpl1 1064	. . . . . . 7 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> A e. GCH )
31		canth2g 8114	. . . . . . 7 |- ( A e. GCH -> A ~< ~P A )
32		sdomdom 7983	. . . . . . 7 |- ( A ~< ~P A -> A ~<_ ~P A )
33	30, 31, 32	3syl 18	. . . . . 6 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> A ~<_ ~P A )
34		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ( A +c A ) ~~ A )
35		pwen 8133	. . . . . . . . 9 |- ( ( A +c A ) ~~ A -> ~P ( A +c A ) ~~ ~P A )
36	34, 35	syl 17	. . . . . . . 8 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ~P ( A +c A ) ~~ ~P A )
37		enen2 8101	. . . . . . . . 9 |- ( ( A +c B ) ~~ ~P A -> ( ~P ( A +c A ) ~~ ( A +c B ) <-> ~P ( A +c A ) ~~ ~P A ) )
38	37	adantl 482	. . . . . . . 8 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ( ~P ( A +c A ) ~~ ( A +c B ) <-> ~P ( A +c A ) ~~ ~P A ) )
39	36, 38	mpbird 247	. . . . . . 7 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ~P ( A +c A ) ~~ ( A +c B ) )
40		endom 7982	. . . . . . 7 |- ( ~P ( A +c A ) ~~ ( A +c B ) -> ~P ( A +c A ) ~<_ ( A +c B ) )
41		pwcdadom 9038	. . . . . . 7 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )
42	39, 40, 41	3syl 18	. . . . . 6 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ~P A ~<_ B )
43		domtr 8009	. . . . . 6 |- ( ( A ~<_ ~P A /\ ~P A ~<_ B ) -> A ~<_ B )
44	33, 42, 43	syl2anc 693	. . . . 5 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> A ~<_ B )
45	44	orcd 407	. . . 4 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A +c B ) ~~ ~P A ) -> ( A ~<_ B \/ B ~<_ A ) )
46	29, 45	jaodan 826	. . 3 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ ( A ~~ ( A +c B ) \/ ( A +c B ) ~~ ~P A ) ) -> ( A ~<_ B \/ B ~<_ A ) )
47	20, 46	syldan 487	. 2 |- ( ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) /\ -. A e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
48	9, 47	pm2.61dan 832	1 |- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A ~<_ B \/ B ~<_ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   +c ccda 8989  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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765  df-cda 8990  df-gch 9443

This theorem is referenced by:  gchaclem  9500