Theorem gchhar 9060

Description: A "local" form of gchac 9062. 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 7990	. . . 4 |- (har ` A ) e. On
2		simp3 999	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. GCH )
3		cdadom3 8571	. . . 4 |- ( ( (har ` A ) e. On /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
4	1, 2, 3	sylancr 663	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
5		domnsym 7645	. . . . . . . . 9 |- ( om ~<_ A -> -. A ~< om )
6	5	3ad2ant1 1018	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~< om )
7		isfinite 8072	. . . . . . . 8 |- ( A e. Fin <-> A ~< om )
8	6, 7	sylnibr 305	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A e. Fin )
9		pwfi 7817	. . . . . . 7 |- ( A e. Fin <-> ~P A e. Fin )
10	8, 9	sylnib 304	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. ~P A e. Fin )
11		cdadom3 8571	. . . . . . 7 |- ( ( ~P A e. GCH /\ (har ` A ) e. On ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
12	2, 1, 11	sylancl 662	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
13		ovex 6309	. . . . . . . 8 |- ( ~P A +c (har ` A ) ) e. _V
14	13	canth2 7672	. . . . . . 7 |- ( ~P A +c (har ` A ) ) ~< ~P ( ~P A +c (har ` A ) )
15		pwcdaen 8568	. . . . . . . . 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 662	. . . . . . . 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 4621	. . . . . . . . . . 11 |- ( ~P A e. GCH -> ~P ~P A e. _V )
18	2, 17	syl 16	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ~P A e. _V )
19		simp2 998	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A e. GCH )
20		harwdom 8019	. . . . . . . . . . 11 |- ( A e. GCH -> (har ` A ) ~<_* ~P ( A X. A ) )
21		wdompwdom 8007	. . . . . . . . . . 11 |- ( (har ` A ) ~<_* ~P ( A X. A ) -> ~P (har ` A ) ~<_ ~P ~P ( A X. A ) )
22	19, 20, 21	3syl 20	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P (har ` A ) ~<_ ~P ~P ( A X. A ) )
23		xpdom2g 7615	. . . . . . . . . 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 661	. . . . . . . . 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 6587	. . . . . . . . . . . . . 14 |- ( ( A e. GCH /\ A e. GCH ) -> ( A X. A ) e. _V )
26	19, 19, 25	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) e. _V )
27		pwexg 4621	. . . . . . . . . . . . 13 |- ( ( A X. A ) e. _V -> ~P ( A X. A ) e. _V )
28	26, 27	syl 16	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A X. A ) e. _V )
29		pwcdaen 8568	. . . . . . . . . . . 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 661	. . . . . . . . . . 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 7568	. . . . . . . . . 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 7549	. . . . . . . . . . . . . 14 |- ( ~P A e. GCH -> ~P A ~~ ~P A )
33	2, 32	syl 16	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ~P A )
34		gchxpidm 9050	. . . . . . . . . . . . . . 15 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~~ A )
35	19, 8, 34	syl2anc 661	. . . . . . . . . . . . . 14 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) ~~ A )
36		pwen 7692	. . . . . . . . . . . . . 14 |- ( ( A X. A ) ~~ A -> ~P ( A X. A ) ~~ ~P A )
37	35, 36	syl 16	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A X. A ) ~~ ~P A )
38		cdaen 8556	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P A +c ~P A ) )
40		gchcdaidm 9049	. . . . . . . . . . . . 13 |- ( ( ~P A e. GCH /\ -. ~P A e. Fin ) -> ( ~P A +c ~P A ) ~~ ~P A )
41	2, 10, 40	syl2anc 661	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P A ) ~~ ~P A )
42		entr 7569	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ~P A )
44		pwen 7692	. . . . . . . . . . 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 16	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c ~P ( A X. A ) ) ~~ ~P ~P A )
46		entr 7569	. . . . . . . . . 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 661	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ~P A )
48		domentr 7576	. . . . . . . . 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 661	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ~P ~P A )
50		endomtr 7575	. . . . . . . 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 661	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c (har ` A ) ) ~<_ ~P ~P A )
52		sdomdomtr 7652	. . . . . . 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 663	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c (har ` A ) ) ~< ~P ~P A )
54		gchen1 9006	. . . . . 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 1230	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( ~P A +c (har ` A ) ) )
56		cdacomen 8564	. . . . 5 |- ( ~P A +c (har ` A ) ) ~~ ( (har ` A ) +c ~P A )
57		entr 7569	. . . . 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 662	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( (har ` A ) +c ~P A ) )
59	58	ensymd 7568	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( (har ` A ) +c ~P A ) ~~ ~P A )
60		domentr 7576	. . 3 |- ( ( (har ` A ) ~<_ ( (har ` A ) +c ~P A ) /\ ( (har ` A ) +c ~P A ) ~~ ~P A ) -> (har ` A ) ~<_ ~P A )
61	4, 59, 60	syl2anc 661	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ~P A )
62		gchcdaidm 9049	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~~ A )
63	19, 8, 62	syl2anc 661	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c A ) ~~ A )
64		pwen 7692	. . . . 5 |- ( ( A +c A ) ~~ A -> ~P ( A +c A ) ~~ ~P A )
65	63, 64	syl 16	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A +c A ) ~~ ~P A )
66		cdadom3 8571	. . . . . . . 8 |- ( ( A e. GCH /\ (har ` A ) e. On ) -> A ~<_ ( A +c (har ` A ) ) )
67	19, 1, 66	sylancl 662	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~<_ ( A +c (har ` A ) ) )
68		harndom 7993	. . . . . . . 8 |- -. (har ` A ) ~<_ A
69		cdadom3 8571	. . . . . . . . . . 11 |- ( ( (har ` A ) e. On /\ A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
70	1, 19, 69	sylancr 663	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
71		cdacomen 8564	. . . . . . . . . 10 |- ( (har ` A ) +c A ) ~~ ( A +c (har ` A ) )
72		domentr 7576	. . . . . . . . . 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 662	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( A +c (har ` A ) ) )
74		domen2 7662	. . . . . . . . 9 |- ( A ~~ ( A +c (har ` A ) ) -> ( (har ` A ) ~<_ A <-> (har ` A ) ~<_ ( A +c (har ` A ) ) ) )
75	73, 74	syl5ibrcom 222	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A ~~ ( A +c (har ` A ) ) -> (har ` A ) ~<_ A ) )
76	68, 75	mtoi 178	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~~ ( A +c (har ` A ) ) )
77		brsdom 7540	. . . . . . 7 |- ( A ~< ( A +c (har ` A ) ) <-> ( A ~<_ ( A +c (har ` A ) ) /\ -. A ~~ ( A +c (har ` A ) ) ) )
78	67, 76, 77	sylanbrc 664	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~< ( A +c (har ` A ) ) )
79		canth2g 7673	. . . . . . . . 9 |- ( A e. GCH -> A ~< ~P A )
80		sdomdom 7545	. . . . . . . . 9 |- ( A ~< ~P A -> A ~<_ ~P A )
81		cdadom1 8569	. . . . . . . . 9 |- ( A ~<_ ~P A -> ( A +c (har ` A ) ) ~<_ ( ~P A +c (har ` A ) ) )
82	19, 79, 80, 81	4syl 21	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c (har ` A ) ) )
83		cdadom2 8570	. . . . . . . . 9 |- ( (har ` A ) ~<_ ~P A -> ( ~P A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
84	61, 83	syl 16	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
85		domtr 7570	. . . . . . . 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 661	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
87		domentr 7576	. . . . . . 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 661	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ~P A )
89		gchen2 9007	. . . . . 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 1230	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~~ ~P A )
91	90	ensymd 7568	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( A +c (har ` A ) ) )
92		entr 7569	. . . 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 661	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A +c A ) ~~ ( A +c (har ` A ) ) )
94		endom 7544	. . 3 |- ( ~P ( A +c A ) ~~ ( A +c (har ` A ) ) -> ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) )
95		pwcdadom 8599	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) -> ~P A ~<_ (har ` A ) )
96	93, 94, 95	3syl 20	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~<_ (har ` A ) )
97		sbth 7639	. 2 |- ( ( (har ` A ) ~<_ ~P A /\ ~P A ~<_ (har ` A ) ) -> (har ` A ) ~~ ~P A )
98	61, 96, 97	syl2anc 661	1 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~~ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 974   e. wcel 1804   _Vcvv 3095   ~Pcpw 3997   class class class wbr 4437   Oncon0 4868   X. cxp 4987   ` cfv 5578  (class class class)co 6281   omcom 6685   ~~ cen 7515   ~<_ cdom 7516   ~< csdm 7517   Fincfn 7518  harchar 7985   ~<_^* cwdom 7986   +c ccda 8550  GCHcgch 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-har 7987  df-wdom 7988  df-cnf 8082  df-card 8323  df-cda 8551  df-fin4 8670  df-gch 9002

This theorem is referenced by:  gchacg  9061