Theorem gchpwdom 9492

Description: A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)

Assertion

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

Proof of Theorem gchpwdom

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> A e. GCH )
2		pwexg 4850	. . . . . . 7 |- ( A e. GCH -> ~P A e. _V )
3	1, 2	syl 17	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ~P A e. _V )
4		simpl3 1066	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> B e. GCH )
5		cdadom3 9010	. . . . . 6 |- ( ( ~P A e. _V /\ B e. GCH ) -> ~P A ~<_ ( ~P A +c B ) )
6	3, 4, 5	syl2anc 693	. . . . 5 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ~P A ~<_ ( ~P A +c B ) )
7		domen2 8103	. . . . 5 |- ( B ~~ ( ~P A +c B ) -> ( ~P A ~<_ B <-> ~P A ~<_ ( ~P A +c B ) ) )
8	6, 7	syl5ibrcom 237	. . . 4 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( B ~~ ( ~P A +c B ) -> ~P A ~<_ B ) )
9		cdacomen 9003	. . . . . . 7 |- ( B +c ~P A ) ~~ ( ~P A +c B )
10		entr 8008	. . . . . . 7 |- ( ( ( B +c ~P A ) ~~ ( ~P A +c B ) /\ ( ~P A +c B ) ~~ ~P B ) -> ( B +c ~P A ) ~~ ~P B )
11	9, 10	mpan 706	. . . . . 6 |- ( ( ~P A +c B ) ~~ ~P B -> ( B +c ~P A ) ~~ ~P B )
12		ensym 8005	. . . . . 6 |- ( ( B +c ~P A ) ~~ ~P B -> ~P B ~~ ( B +c ~P A ) )
13		endom 7982	. . . . . 6 |- ( ~P B ~~ ( B +c ~P A ) -> ~P B ~<_ ( B +c ~P A ) )
14	11, 12, 13	3syl 18	. . . . 5 |- ( ( ~P A +c B ) ~~ ~P B -> ~P B ~<_ ( B +c ~P A ) )
15		domsdomtr 8095	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A ~< B ) -> om ~< B )
16	15	3ad2antl1 1223	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> om ~< B )
17		sdomnsym 8085	. . . . . . . . . 10 |- ( om ~< B -> -. B ~< om )
18	16, 17	syl 17	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> -. B ~< om )
19		isfinite 8549	. . . . . . . . 9 |- ( B e. Fin <-> B ~< om )
20	18, 19	sylnibr 319	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> -. B e. Fin )
21		gchcdaidm 9490	. . . . . . . 8 |- ( ( B e. GCH /\ -. B e. Fin ) -> ( B +c B ) ~~ B )
22	4, 20, 21	syl2anc 693	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( B +c B ) ~~ B )
23		pwen 8133	. . . . . . 7 |- ( ( B +c B ) ~~ B -> ~P ( B +c B ) ~~ ~P B )
24		domen1 8102	. . . . . . 7 |- ( ~P ( B +c B ) ~~ ~P B -> ( ~P ( B +c B ) ~<_ ( B +c ~P A ) <-> ~P B ~<_ ( B +c ~P A ) ) )
25	22, 23, 24	3syl 18	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P ( B +c B ) ~<_ ( B +c ~P A ) <-> ~P B ~<_ ( B +c ~P A ) ) )
26		pwcdadom 9038	. . . . . . 7 |- ( ~P ( B +c B ) ~<_ ( B +c ~P A ) -> ~P B ~<_ ~P A )
27		canth2g 8114	. . . . . . . . 9 |- ( B e. GCH -> B ~< ~P B )
28		sdomdomtr 8093	. . . . . . . . . 10 |- ( ( B ~< ~P B /\ ~P B ~<_ ~P A ) -> B ~< ~P A )
29	28	ex 450	. . . . . . . . 9 |- ( B ~< ~P B -> ( ~P B ~<_ ~P A -> B ~< ~P A ) )
30	4, 27, 29	3syl 18	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B ~<_ ~P A -> B ~< ~P A ) )
31		gchi 9446	. . . . . . . . . 10 |- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )
32	31	3expia 1267	. . . . . . . . 9 |- ( ( A e. GCH /\ A ~< B ) -> ( B ~< ~P A -> A e. Fin ) )
33	32	3ad2antl2 1224	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( B ~< ~P A -> A e. Fin ) )
34		isfinite 8549	. . . . . . . . 9 |- ( A e. Fin <-> A ~< om )
35		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> om ~<_ A )
36		domnsym 8086	. . . . . . . . . . 11 |- ( om ~<_ A -> -. A ~< om )
37	35, 36	syl 17	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> -. A ~< om )
38	37	pm2.21d 118	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( A ~< om -> ~P A ~<_ B ) )
39	34, 38	syl5bi 232	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( A e. Fin -> ~P A ~<_ B ) )
40	30, 33, 39	3syld 60	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B ~<_ ~P A -> ~P A ~<_ B ) )
41	26, 40	syl5 34	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P ( B +c B ) ~<_ ( B +c ~P A ) -> ~P A ~<_ B ) )
42	25, 41	sylbird 250	. . . . 5 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B ~<_ ( B +c ~P A ) -> ~P A ~<_ B ) )
43	14, 42	syl5 34	. . . 4 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ( ~P A +c B ) ~~ ~P B -> ~P A ~<_ B ) )
44		cdadom3 9010	. . . . . . 7 |- ( ( B e. GCH /\ ~P A e. _V ) -> B ~<_ ( B +c ~P A ) )
45	4, 3, 44	syl2anc 693	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> B ~<_ ( B +c ~P A ) )
46		domentr 8015	. . . . . 6 |- ( ( B ~<_ ( B +c ~P A ) /\ ( B +c ~P A ) ~~ ( ~P A +c B ) ) -> B ~<_ ( ~P A +c B ) )
47	45, 9, 46	sylancl 694	. . . . 5 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> B ~<_ ( ~P A +c B ) )
48		sdomdom 7983	. . . . . . . . 9 |- ( A ~< B -> A ~<_ B )
49	48	adantl 482	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> A ~<_ B )
50		pwdom 8112	. . . . . . . 8 |- ( A ~<_ B -> ~P A ~<_ ~P B )
51		cdadom1 9008	. . . . . . . 8 |- ( ~P A ~<_ ~P B -> ( ~P A +c B ) ~<_ ( ~P B +c B ) )
52	49, 50, 51	3syl 18	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P A +c B ) ~<_ ( ~P B +c B ) )
53	4, 27	syl 17	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> B ~< ~P B )
54		sdomdom 7983	. . . . . . . 8 |- ( B ~< ~P B -> B ~<_ ~P B )
55		cdadom2 9009	. . . . . . . 8 |- ( B ~<_ ~P B -> ( ~P B +c B ) ~<_ ( ~P B +c ~P B ) )
56	53, 54, 55	3syl 18	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B +c B ) ~<_ ( ~P B +c ~P B ) )
57		domtr 8009	. . . . . . 7 |- ( ( ( ~P A +c B ) ~<_ ( ~P B +c B ) /\ ( ~P B +c B ) ~<_ ( ~P B +c ~P B ) ) -> ( ~P A +c B ) ~<_ ( ~P B +c ~P B ) )
58	52, 56, 57	syl2anc 693	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P A +c B ) ~<_ ( ~P B +c ~P B ) )
59		pwcda1 9016	. . . . . . . 8 |- ( B e. GCH -> ( ~P B +c ~P B ) ~~ ~P ( B +c 1o ) )
60	4, 59	syl 17	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B +c ~P B ) ~~ ~P ( B +c 1o ) )
61		gchcda1 9478	. . . . . . . . 9 |- ( ( B e. GCH /\ -. B e. Fin ) -> ( B +c 1o ) ~~ B )
62	4, 20, 61	syl2anc 693	. . . . . . . 8 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( B +c 1o ) ~~ B )
63		pwen 8133	. . . . . . . 8 |- ( ( B +c 1o ) ~~ B -> ~P ( B +c 1o ) ~~ ~P B )
64	62, 63	syl 17	. . . . . . 7 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ~P ( B +c 1o ) ~~ ~P B )
65		entr 8008	. . . . . . 7 |- ( ( ( ~P B +c ~P B ) ~~ ~P ( B +c 1o ) /\ ~P ( B +c 1o ) ~~ ~P B ) -> ( ~P B +c ~P B ) ~~ ~P B )
66	60, 64, 65	syl2anc 693	. . . . . 6 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P B +c ~P B ) ~~ ~P B )
67		domentr 8015	. . . . . 6 |- ( ( ( ~P A +c B ) ~<_ ( ~P B +c ~P B ) /\ ( ~P B +c ~P B ) ~~ ~P B ) -> ( ~P A +c B ) ~<_ ~P B )
68	58, 66, 67	syl2anc 693	. . . . 5 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( ~P A +c B ) ~<_ ~P B )
69		gchor 9449	. . . . 5 |- ( ( ( B e. GCH /\ -. B e. Fin ) /\ ( B ~<_ ( ~P A +c B ) /\ ( ~P A +c B ) ~<_ ~P B ) ) -> ( B ~~ ( ~P A +c B ) \/ ( ~P A +c B ) ~~ ~P B ) )
70	4, 20, 47, 68, 69	syl22anc 1327	. . . 4 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ( B ~~ ( ~P A +c B ) \/ ( ~P A +c B ) ~~ ~P B ) )
71	8, 43, 70	mpjaod 396	. . 3 |- ( ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) /\ A ~< B ) -> ~P A ~<_ B )
72	71	ex 450	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) -> ( A ~< B -> ~P A ~<_ B ) )
73		reldom 7961	. . . . 5 |- Rel ~<_
74	73	brrelexi 5158	. . . 4 |- ( ~P A ~<_ B -> ~P A e. _V )
75		pwexb 6975	. . . . 5 |- ( A e. _V <-> ~P A e. _V )
76		canth2g 8114	. . . . 5 |- ( A e. _V -> A ~< ~P A )
77	75, 76	sylbir 225	. . . 4 |- ( ~P A e. _V -> A ~< ~P A )
78	74, 77	syl 17	. . 3 |- ( ~P A ~<_ B -> A ~< ~P A )
79		sdomdomtr 8093	. . 3 |- ( ( A ~< ~P A /\ ~P A ~<_ B ) -> A ~< B )
80	78, 79	mpancom 703	. 2 |- ( ~P A ~<_ B -> A ~< B )
81	72, 80	impbid1 215	1 |- ( ( om ~<_ A /\ A e. GCH /\ B e. GCH ) -> ( A ~< B <-> ~P A ~<_ B ) )

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   omcom 7065   1oc1o 7553   ~~ 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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rmo 2920  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-se 5074  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-pred 5680  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-wdom 8464  df-cnf 8559  df-card 8765  df-cda 8990  df-fin4 9109  df-gch 9443

This theorem is referenced by:  gchaleph2  9494  gchina  9521