Theorem gchhar 9109

Description: A "local" form of gchac 9111. 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 8081	. . . 4 |- (har ` A ) e. On
2		simp3 1011	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A e. GCH )
3		cdadom3 8623	. . . 4 |- ( ( (har ` A ) e. On /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
4	1, 2, 3	sylancr 670	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c ~P A ) )
5		domnsym 7703	. . . . . . . . 9 |- ( om ~<_ A -> -. A ~< om )
6	5	3ad2ant1 1030	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~< om )
7		isfinite 8162	. . . . . . . 8 |- ( A e. Fin <-> A ~< om )
8	6, 7	sylnibr 307	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A e. Fin )
9		pwfi 7874	. . . . . . 7 |- ( A e. Fin <-> ~P A e. Fin )
10	8, 9	sylnib 306	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. ~P A e. Fin )
11		cdadom3 8623	. . . . . . 7 |- ( ( ~P A e. GCH /\ (har ` A ) e. On ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
12	2, 1, 11	sylancl 669	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~<_ ( ~P A +c (har ` A ) ) )
13		ovex 6323	. . . . . . . 8 |- ( ~P A +c (har ` A ) ) e. _V
14	13	canth2 7730	. . . . . . 7 |- ( ~P A +c (har ` A ) ) ~< ~P ( ~P A +c (har ` A ) )
15		pwcdaen 8620	. . . . . . . . 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 669	. . . . . . . 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 4590	. . . . . . . . . . 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 1010	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A e. GCH )
20		harwdom 8110	. . . . . . . . . . 11 |- ( A e. GCH -> (har ` A ) ~<_* ~P ( A X. A ) )
21		wdompwdom 8098	. . . . . . . . . . 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 7673	. . . . . . . . . 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 667	. . . . . . . . 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 6598	. . . . . . . . . . . . . 14 |- ( ( A e. GCH /\ A e. GCH ) -> ( A X. A ) e. _V )
26	19, 19, 25	syl2anc 667	. . . . . . . . . . . . 13 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) e. _V )
27		pwexg 4590	. . . . . . . . . . . . 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 8620	. . . . . . . . . . . 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 667	. . . . . . . . . . 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 7625	. . . . . . . . . 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 7606	. . . . . . . . . . . . . 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 9099	. . . . . . . . . . . . . . 15 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~~ A )
35	19, 8, 34	syl2anc 667	. . . . . . . . . . . . . 14 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A X. A ) ~~ A )
36		pwen 7750	. . . . . . . . . . . . . 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 8608	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ( ~P A +c ~P A ) )
40		gchcdaidm 9098	. . . . . . . . . . . . 13 |- ( ( ~P A e. GCH /\ -. ~P A e. Fin ) -> ( ~P A +c ~P A ) ~~ ~P A )
41	2, 10, 40	syl2anc 667	. . . . . . . . . . . 12 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P A ) ~~ ~P A )
42		entr 7626	. . . . . . . . . . . 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 667	. . . . . . . . . . 11 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c ~P ( A X. A ) ) ~~ ~P A )
44		pwen 7750	. . . . . . . . . . 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 7626	. . . . . . . . . 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 667	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P ~P ( A X. A ) ) ~~ ~P ~P A )
48		domentr 7633	. . . . . . . . 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 667	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P ~P A X. ~P (har ` A ) ) ~<_ ~P ~P A )
50		endomtr 7632	. . . . . . . 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 667	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( ~P A +c (har ` A ) ) ~<_ ~P ~P A )
52		sdomdomtr 7710	. . . . . . 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 670	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( ~P A +c (har ` A ) ) ~< ~P ~P A )
54		gchen1 9055	. . . . . 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 1270	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( ~P A +c (har ` A ) ) )
56		cdacomen 8616	. . . . 5 |- ( ~P A +c (har ` A ) ) ~~ ( (har ` A ) +c ~P A )
57		entr 7626	. . . . 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 669	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( (har ` A ) +c ~P A ) )
59	58	ensymd 7625	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( (har ` A ) +c ~P A ) ~~ ~P A )
60		domentr 7633	. . 3 |- ( ( (har ` A ) ~<_ ( (har ` A ) +c ~P A ) /\ ( (har ` A ) +c ~P A ) ~~ ~P A ) -> (har ` A ) ~<_ ~P A )
61	4, 59, 60	syl2anc 667	. 2 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ~P A )
62		gchcdaidm 9098	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~~ A )
63	19, 8, 62	syl2anc 667	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c A ) ~~ A )
64		pwen 7750	. . . . 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 8623	. . . . . . . 8 |- ( ( A e. GCH /\ (har ` A ) e. On ) -> A ~<_ ( A +c (har ` A ) ) )
67	19, 1, 66	sylancl 669	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~<_ ( A +c (har ` A ) ) )
68		harndom 8084	. . . . . . . 8 |- -. (har ` A ) ~<_ A
69		cdadom3 8623	. . . . . . . . . . 11 |- ( ( (har ` A ) e. On /\ A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
70	1, 19, 69	sylancr 670	. . . . . . . . . 10 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( (har ` A ) +c A ) )
71		cdacomen 8616	. . . . . . . . . 10 |- ( (har ` A ) +c A ) ~~ ( A +c (har ` A ) )
72		domentr 7633	. . . . . . . . . 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 669	. . . . . . . . 9 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~<_ ( A +c (har ` A ) ) )
74		domen2 7720	. . . . . . . . 9 |- ( A ~~ ( A +c (har ` A ) ) -> ( (har ` A ) ~<_ A <-> (har ` A ) ~<_ ( A +c (har ` A ) ) ) )
75	73, 74	syl5ibrcom 226	. . . . . . . 8 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A ~~ ( A +c (har ` A ) ) -> (har ` A ) ~<_ A ) )
76	68, 75	mtoi 182	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> -. A ~~ ( A +c (har ` A ) ) )
77		brsdom 7597	. . . . . . 7 |- ( A ~< ( A +c (har ` A ) ) <-> ( A ~<_ ( A +c (har ` A ) ) /\ -. A ~~ ( A +c (har ` A ) ) ) )
78	67, 76, 77	sylanbrc 671	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> A ~< ( A +c (har ` A ) ) )
79		canth2g 7731	. . . . . . . . 9 |- ( A e. GCH -> A ~< ~P A )
80		sdomdom 7602	. . . . . . . . 9 |- ( A ~< ~P A -> A ~<_ ~P A )
81		cdadom1 8621	. . . . . . . . 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 8622	. . . . . . . . 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 7627	. . . . . . . 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 667	. . . . . . 7 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ( ~P A +c ~P A ) )
87		domentr 7633	. . . . . . 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 667	. . . . . 6 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~<_ ~P A )
89		gchen2 9056	. . . . . 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 1270	. . . . 5 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ( A +c (har ` A ) ) ~~ ~P A )
91	90	ensymd 7625	. . . 4 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P A ~~ ( A +c (har ` A ) ) )
92		entr 7626	. . . 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 667	. . 3 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> ~P ( A +c A ) ~~ ( A +c (har ` A ) ) )
94		endom 7601	. . 3 |- ( ~P ( A +c A ) ~~ ( A +c (har ` A ) ) -> ~P ( A +c A ) ~<_ ( A +c (har ` A ) ) )
95		pwcdadom 8651	. . 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 7697	. 2 |- ( ( (har ` A ) ~<_ ~P A /\ ~P A ~<_ (har ` A ) ) -> (har ` A ) ~~ ~P A )
98	61, 96, 97	syl2anc 667	1 |- ( ( om ~<_ A /\ A e. GCH /\ ~P A e. GCH ) -> (har ` A ) ~~ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 986   e. wcel 1889   _Vcvv 3047   ~Pcpw 3953   class class class wbr 4405   X. cxp 4835   Oncon0 5426   ` cfv 5585  (class class class)co 6295   omcom 6697   ~~ cen 7571   ~<_ cdom 7572   ~< csdm 7573   Fincfn 7574  harchar 8076   ~<_^* cwdom 8077   +c ccda 8602  GCHcgch 9050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-seqom 7170  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-oexp 7193  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-oi 8030  df-har 8078  df-wdom 8079  df-cnf 8172  df-card 8378  df-cda 8603  df-fin4 8722  df-gch 9051

This theorem is referenced by:  gchacg  9110