Theorem gchhar 8851

Description: A "local" form of gchac 8853. 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 7781	. . . 4 |- (har ` A ) e. On
2		simp3 990	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. GCH )
3		cdadom3 8362	. . . 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 7442	. . . . . . . . 9 |- ( om ~<_ A -> -. A ~< om )
6	5	3ad2ant1 1009	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~< om )
7		isfinite 7863	. . . . . . . 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 7611	. . . . . . 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 8362	. . . . . . 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 6121	. . . . . . . 8 |- ( ~P A +c (har ` A ) ) e. _V
14	13	canth2 7469	. . . . . . 7 |- ( ~P A +c (har ` A ) ) ~< ~P ( ~P A +c (har ` A ) )
15		pwcdaen 8359	. . . . . . . . 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 4481	. . . . . . . . . . 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 989	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A e. GCH )
20		harwdom 7810	. . . . . . . . . . 11 |- ( A e. GCH -> (har ` A ) ~<_* ~P ( A X. A ) )
21		wdompwdom 7798	. . . . . . . . . . 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 7412	. . . . . . . . . 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 6512	. . . . . . . . . . . . . 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 4481	. . . . . . . . . . . . 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 8359	. . . . . . . . . . . 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 7365	. . . . . . . . . 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 7346	. . . . . . . . . . . . . 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 8841	. . . . . . . . . . . . . . 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 7489	. . . . . . . . . . . . . 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 8347	. . . . . . . . . . . . 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 8840	. . . . . . . . . . . . 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 7366	. . . . . . . . . . . 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 7489	. . . . . . . . . . 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 7366	. . . . . . . . . 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 7373	. . . . . . . . 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 7372	. . . . . . . 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 7449	. . . . . . 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 8797	. . . . . 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 1219	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( ~P A +c (har ` A ) ) )
56		cdacomen 8355	. . . . 5 |- ( ~P A +c (har ` A ) ) ~~ ( (har ` A ) +c ~P A )
57		entr 7366	. . . . 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 7365	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( (har ` A ) +c ~P A ) ~~ ~P A )
60		domentr 7373	. . 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 8840	. . . . . 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 7489	. . . . 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 8362	. . . . . . . 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 7784	. . . . . . . 8 |- -. (har ` A ) ~<_ A
69		cdadom3 8362	. . . . . . . . . . 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 8355	. . . . . . . . . 10 |- ( (har ` A ) +c A ) ~~ ( A +c (har ` A ) )
72		domentr 7373	. . . . . . . . . 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 7459	. . . . . . . . 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 7337	. . . . . . 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 7470	. . . . . . . . 9 |- ( A e. GCH -> A ~< ~P A )
80		sdomdom 7342	. . . . . . . . 9 |- ( A ~< ~P A -> A ~<_ ~P A )
81		cdadom1 8360	. . . . . . . . 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 8361	. . . . . . . . 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 7367	. . . . . . . 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 7373	. . . . . . 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 8798	. . . . . 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 1219	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~~ ~P A )
91	90	ensymd 7365	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( A +c (har ` A ) ) )
92		entr 7366	. . . 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 7341	. . 3 |- ( ~P ( A +c A ) ~~ ( A +c (har ` A ) ) -> ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) )
95		pwcdadom 8390	. . 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 7436	. 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 965   e. wcel 1756   _Vcvv 2977   ~Pcpw 3865   class class class wbr 4297   Oncon0 4724   X. cxp 4843   ` cfv 5423  (class class class)co 6096   omcom 6481   ~~ cen 7312   ~<_ cdom 7313   ~< csdm 7314   Fincfn 7315  harchar 7776   ~<_^* cwdom 7777   +c ccda 8341  GCHcgch 8792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-oexp 6931  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-har 7778  df-wdom 7779  df-cnf 7873  df-card 8114  df-cda 8342  df-fin4 8461  df-gch 8793

This theorem is referenced by:  gchacg  8852