Theorem gchhar 9501

Description: A "local" form of gchac 9503. If A and ~P A are GCH-sets, then the Hartogs number of A is ~P A (so ~P A and a fortiori A are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)

Assertion

Ref	Expression
gchhar	|- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~~ ~P A )

Proof of Theorem gchhar

Step	Hyp	Ref	Expression
1		harcl 8466	. . . 4 |- (har ` A ) e. On
2		simp3 1063	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. GCH )
3		cdadom3 9010	. . . 4 |- ( ( (har ` A ) e. On /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
4	1, 2, 3	sylancr 695	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
5		domnsym 8086	. . . . . . . . 9 |- ( om ~<_ A -> -. A ~< om )
6	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~< om )
7		isfinite 8549	. . . . . . . 8 |- ( A e. Fin <-> A ~< om )
8	6, 7	sylnibr 319	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A e. Fin )
9		pwfi 8261	. . . . . . 7 |- ( A e. Fin <-> ~P A e. Fin )
10	8, 9	sylnib 318	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. ~P A e. Fin )
11		cdadom3 9010	. . . . . . 7 |- ( ( ~P A e. GCH /\ (har ` A ) e. On ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
12	2, 1, 11	sylancl 694	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
13		ovex 6678	. . . . . . . 8 |- ( ~P A +c (har ` A ) ) e. _V
14	13	canth2 8113	. . . . . . 7 |- ( ~P A +c (har ` A ) ) ~< ~P ( ~P A +c (har ` A ) )
15		pwcdaen 9007	. . . . . . . . 9 |- ( ( ~P A e. GCH /\ (har ` A ) e. On ) -> ~P ( ~P A +c (har ` A ) ) ~~ ( ~P ~P A X. ~P (har ` A ) ) )
16	2, 1, 15	sylancl 694	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c (har ` A ) ) ~~ ( ~P ~P A X. ~P (har ` A ) ) )
17		pwexg 4850	. . . . . . . . . . 11 |- ( ~P A e. GCH -> ~P ~P A e. _V )
18	2, 17	syl 17	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ~P A e. _V )
19		simp2 1062	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A e. GCH )
20		harwdom 8495	. . . . . . . . . . 11 |- ( A e. GCH -> (har ` A ) ~<_* ~P ( A X. A ) )
21		wdompwdom 8483	. . . . . . . . . . 11 |- ( (har ` A ) ~<_* ~P ( A X. A ) -> ~P (har ` A ) ~<_ ~P ~P ( A X. A ) )
22	19, 20, 21	3syl 18	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P (har ` A ) ~<_ ~P ~P ( A X. A ) )
23		xpdom2g 8056	. . . . . . . . . 10 |- ( ( ~P ~P A e. _V /\ ~P (har ` A ) ~<_ ~P ~P ( A X. A ) ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ( ~P ~P A X. ~P ~P ( A X. A ) ) )
24	18, 22, 23	syl2anc 693	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ( ~P ~P A X. ~P ~P ( A X. A ) ) )
25		xpexg 6960	. . . . . . . . . . . . . 14 |- ( ( A e. GCH /\ A e. GCH ) -> ( A X. A ) e. _V )
26	19, 19, 25	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) e. _V )
27		pwexg 4850	. . . . . . . . . . . . 13 |- ( ( A X. A ) e. _V -> ~P ( A X. A ) e. _V )
28	26, 27	syl 17	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A X. A ) e. _V )
29		pwcdaen 9007	. . . . . . . . . . . 12 |- ( ( ~P A e. GCH /\ ~P ( A X. A ) e. _V ) -> ~P ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P ~P A X. ~P ~P ( A X. A ) ) )
30	2, 28, 29	syl2anc 693	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P ~P A X. ~P ~P ( A X. A ) ) )
31	30	ensymd 8007	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ( ~P A +c ~P ( A X. A ) ) )
32		enrefg 7987	. . . . . . . . . . . . . 14 |- ( ~P A e. GCH -> ~P A ~~ ~P A )
33	2, 32	syl 17	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ~P A )
34		gchxpidm 9491	. . . . . . . . . . . . . . 15 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~~ A )
35	19, 8, 34	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) ~~ A )
36		pwen 8133	. . . . . . . . . . . . . 14 |- ( ( A X. A ) ~~ A -> ~P ( A X. A ) ~~ ~P A )
37	35, 36	syl 17	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A X. A ) ~~ ~P A )
38		cdaen 8995	. . . . . . . . . . . . 13 |- ( ( ~P A ~~ ~P A /\ ~P ( A X. A ) ~~ ~P A ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P A +c ~P A ) )
39	33, 37, 38	syl2anc 693	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P A +c ~P A ) )
40		gchcdaidm 9490	. . . . . . . . . . . . 13 |- ( ( ~P A e. GCH /\ -. ~P A e. Fin ) -> ( ~P A +c ~P A ) ~~ ~P A )
41	2, 10, 40	syl2anc 693	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P A ) ~~ ~P A )
42		entr 8008	. . . . . . . . . . . 12 |- ( ( ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P A +c ~P A ) /\ ( ~P A +c ~P A ) ~~ ~P A ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ~P A )
43	39, 41, 42	syl2anc 693	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ~P A )
44		pwen 8133	. . . . . . . . . . 11 |- ( ( ~P A +c ~P ( A X. A ) ) ~~ ~P A -> ~P ( ~P A +c ~P ( A X. A ) ) ~~ ~P ~P A )
45	43, 44	syl 17	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c ~P ( A X. A ) ) ~~ ~P ~P A )
46		entr 8008	. . . . . . . . . 10 |- ( ( ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ( ~P A +c ~P ( A X. A ) ) /\ ~P ( ~P A +c ~P ( A X. A ) ) ~~ ~P ~P A ) -> ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ~P A )
47	31, 45, 46	syl2anc 693	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ~P A )
48		domentr 8015	. . . . . . . . 9 |- ( ( ( ~P ~P A X. ~P (har ` A ) ) ~<_ ( ~P ~P A X. ~P ~P ( A X. A ) ) /\ ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ~P A ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ~P ~P A )
49	24, 47, 48	syl2anc 693	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ~P ~P A )
50		endomtr 8014	. . . . . . . 8 |- ( ( ~P ( ~P A +c (har ` A ) ) ~~ ( ~P ~P A X. ~P (har ` A ) ) /\ ( ~P ~P A X. ~P (har ` A ) ) ~<_ ~P ~P A ) -> ~P ( ~P A +c (har ` A ) ) ~<_ ~P ~P A )
51	16, 49, 50	syl2anc 693	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c (har ` A ) ) ~<_ ~P ~P A )
52		sdomdomtr 8093	. . . . . . 7 |- ( ( ( ~P A +c (har ` A ) ) ~< ~P ( ~P A +c (har ` A ) ) /\ ~P ( ~P A +c (har ` A ) ) ~<_ ~P ~P A ) -> ( ~P A +c (har ` A ) ) ~< ~P ~P A )
53	14, 51, 52	sylancr 695	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c (har ` A ) ) ~< ~P ~P A )
54		gchen1 9447	. . . . . 6 |- ( ( ( ~P A e. GCH /\ -. ~P A e. Fin ) /\ ( ~P A ~<_ ( ~P A +c (har ` A ) ) /\ ( ~P A +c (har ` A ) ) ~< ~P ~P A ) ) -> ~P A ~~ ( ~P A +c (har ` A ) ) )
55	2, 10, 12, 53, 54	syl22anc 1327	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( ~P A +c (har ` A ) ) )
56		cdacomen 9003	. . . . 5 |- ( ~P A +c (har ` A ) ) ~~ ( (har ` A ) +c ~P A )
57		entr 8008	. . . . 5 |- ( ( ~P A ~~ ( ~P A +c (har ` A ) ) /\ ( ~P A +c (har ` A ) ) ~~ ( (har ` A ) +c ~P A ) ) -> ~P A ~~ ( (har ` A ) +c ~P A ) )
58	55, 56, 57	sylancl 694	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( (har ` A ) +c ~P A ) )
59	58	ensymd 8007	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( (har ` A ) +c ~P A ) ~~ ~P A )
60		domentr 8015	. . 3 |- ( ( (har ` A ) ~<_ ( (har ` A ) +c ~P A ) /\ ( (har ` A ) +c ~P A ) ~~ ~P A ) -> (har ` A ) ~<_ ~P A )
61	4, 59, 60	syl2anc 693	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ~P A )
62		gchcdaidm 9490	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~~ A )
63	19, 8, 62	syl2anc 693	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c A ) ~~ A )
64		pwen 8133	. . . . 5 |- ( ( A +c A ) ~~ A -> ~P ( A +c A ) ~~ ~P A )
65	63, 64	syl 17	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A +c A ) ~~ ~P A )
66		cdadom3 9010	. . . . . . . 8 |- ( ( A e. GCH /\ (har ` A ) e. On ) -> A ~<_ ( A +c (har ` A ) ) )
67	19, 1, 66	sylancl 694	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~<_ ( A +c (har ` A ) ) )
68		harndom 8469	. . . . . . . 8 |- -. (har ` A ) ~<_ A
69		cdadom3 9010	. . . . . . . . . . 11 |- ( ( (har ` A ) e. On /\ A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
70	1, 19, 69	sylancr 695	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
71		cdacomen 9003	. . . . . . . . . 10 |- ( (har ` A ) +c A ) ~~ ( A +c (har ` A ) )
72		domentr 8015	. . . . . . . . . 10 |- ( ( (har ` A ) ~<_ ( (har ` A ) +c A ) /\ ( (har ` A ) +c A ) ~~ ( A +c (har ` A ) ) ) -> (har ` A ) ~<_ ( A +c (har ` A ) ) )
73	70, 71, 72	sylancl 694	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( A +c (har ` A ) ) )
74		domen2 8103	. . . . . . . . 9 |- ( A ~~ ( A +c (har ` A ) ) -> ( (har ` A ) ~<_ A <-> (har ` A ) ~<_ ( A +c (har ` A ) ) ) )
75	73, 74	syl5ibrcom 237	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A ~~ ( A +c (har ` A ) ) -> (har ` A ) ~<_ A ) )
76	68, 75	mtoi 190	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~~ ( A +c (har ` A ) ) )
77		brsdom 7978	. . . . . . 7 |- ( A ~< ( A +c (har ` A ) ) <-> ( A ~<_ ( A +c (har ` A ) ) /\ -. A ~~ ( A +c (har ` A ) ) ) )
78	67, 76, 77	sylanbrc 698	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~< ( A +c (har ` A ) ) )
79		canth2g 8114	. . . . . . . . 9 |- ( A e. GCH -> A ~< ~P A )
80		sdomdom 7983	. . . . . . . . 9 |- ( A ~< ~P A -> A ~<_ ~P A )
81		cdadom1 9008	. . . . . . . . 9 |- ( A ~<_ ~P A -> ( A +c (har ` A ) ) ~<_ ( ~P A +c (har ` A ) ) )
82	19, 79, 80, 81	4syl 19	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c (har ` A ) ) )
83		cdadom2 9009	. . . . . . . . 9 |- ( (har ` A ) ~<_ ~P A -> ( ~P A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
84	61, 83	syl 17	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
85		domtr 8009	. . . . . . . 8 |- ( ( ( A +c (har ` A ) ) ~<_ ( ~P A +c (har ` A ) ) /\ ( ~P A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
86	82, 84, 85	syl2anc 693	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
87		domentr 8015	. . . . . . 7 |- ( ( ( A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) /\ ( ~P A +c ~P A ) ~~ ~P A ) -> ( A +c (har ` A ) ) ~<_ ~P A )
88	86, 41, 87	syl2anc 693	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ~P A )
89		gchen2 9448	. . . . . 6 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~< ( A +c (har ` A ) ) /\ ( A +c (har ` A ) ) ~<_ ~P A ) ) -> ( A +c (har ` A ) ) ~~ ~P A )
90	19, 8, 78, 88, 89	syl22anc 1327	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~~ ~P A )
91	90	ensymd 8007	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( A +c (har ` A ) ) )
92		entr 8008	. . . 4 |- ( ( ~P ( A +c A ) ~~ ~P A /\ ~P A ~~ ( A +c (har ` A ) ) ) -> ~P ( A +c A ) ~~ ( A +c (har ` A ) ) )
93	65, 91, 92	syl2anc 693	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A +c A ) ~~ ( A +c (har ` A ) ) )
94		endom 7982	. . 3 |- ( ~P ( A +c A ) ~~ ( A +c (har ` A ) ) -> ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) )
95		pwcdadom 9038	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) -> ~P A ~<_ (har ` A ) )
96	93, 94, 95	3syl 18	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~<_ (har ` A ) )
97		sbth 8080	. 2 |- ( ( (har ` A ) ~<_ ~P A /\ ~P A ~<_ (har ` A ) ) -> (har ` A ) ~~ ~P A )
98	61, 96, 97	syl2anc 693	1 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~~ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112   Oncon0 5723   ` cfv 5888  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955  harchar 8461   ~<_^* cwdom 8462   +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:  gchacg  9502