Theorem r1tskina 9604

Description: There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)

Assertion

Ref	Expression
r1tskina	|- ( A e. On -> ( ( R1 ` A ) e. Tarski <-> ( A = (/) \/ A e. Inacc ) ) )

Proof of Theorem r1tskina

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . 5 |- ( A =/= (/) <-> -. A = (/) )
2		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( R1 ` A ) e. Tarski )
3		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> A e. On )
4		onwf 8693	. . . . . . . . . . . . . . . 16 |- On C_ U. ( R1 " On )
5	4	sseli 3599	. . . . . . . . . . . . . . 15 |- ( A e. On -> A e. U. ( R1 " On ) )
6		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( rank ` A ) = ( rank ` A )
7		rankr1c 8684	. . . . . . . . . . . . . . . 16 |- ( A e. U. ( R1 " On ) -> ( ( rank ` A ) = ( rank ` A ) <-> ( -. A e. ( R1 ` ( rank ` A ) ) /\ A e. ( R1 ` suc ( rank ` A ) ) ) ) )
8	6, 7	mpbii 223	. . . . . . . . . . . . . . 15 |- ( A e. U. ( R1 " On ) -> ( -. A e. ( R1 ` ( rank ` A ) ) /\ A e. ( R1 ` suc ( rank ` A ) ) ) )
9	5, 8	syl 17	. . . . . . . . . . . . . 14 |- ( A e. On -> ( -. A e. ( R1 ` ( rank ` A ) ) /\ A e. ( R1 ` suc ( rank ` A ) ) ) )
10	9	simpld 475	. . . . . . . . . . . . 13 |- ( A e. On -> -. A e. ( R1 ` ( rank ` A ) ) )
11		r1fnon 8630	. . . . . . . . . . . . . . . . 17 |- R1 Fn On
12		fndm 5990	. . . . . . . . . . . . . . . . 17 |- ( R1 Fn On -> dom R1 = On )
13	11, 12	ax-mp 5	. . . . . . . . . . . . . . . 16 |- dom R1 = On
14	13	eleq2i 2693	. . . . . . . . . . . . . . 15 |- ( A e. dom R1 <-> A e. On )
15		rankonid 8692	. . . . . . . . . . . . . . 15 |- ( A e. dom R1 <-> ( rank ` A ) = A )
16	14, 15	bitr3i 266	. . . . . . . . . . . . . 14 |- ( A e. On <-> ( rank ` A ) = A )
17		fveq2 6191	. . . . . . . . . . . . . 14 |- ( ( rank ` A ) = A -> ( R1 ` ( rank ` A ) ) = ( R1 ` A ) )
18	16, 17	sylbi 207	. . . . . . . . . . . . 13 |- ( A e. On -> ( R1 ` ( rank ` A ) ) = ( R1 ` A ) )
19	10, 18	neleqtrd 2722	. . . . . . . . . . . 12 |- ( A e. On -> -. A e. ( R1 ` A ) )
20	19	adantl 482	. . . . . . . . . . 11 |- ( ( ( R1 ` A ) e. Tarski /\ A e. On ) -> -. A e. ( R1 ` A ) )
21		onssr1 8694	. . . . . . . . . . . . . 14 |- ( A e. dom R1 -> A C_ ( R1 ` A ) )
22	14, 21	sylbir 225	. . . . . . . . . . . . 13 |- ( A e. On -> A C_ ( R1 ` A ) )
23		tsken 9576	. . . . . . . . . . . . 13 |- ( ( ( R1 ` A ) e. Tarski /\ A C_ ( R1 ` A ) ) -> ( A ~~ ( R1 ` A ) \/ A e. ( R1 ` A ) ) )
24	22, 23	sylan2 491	. . . . . . . . . . . 12 |- ( ( ( R1 ` A ) e. Tarski /\ A e. On ) -> ( A ~~ ( R1 ` A ) \/ A e. ( R1 ` A ) ) )
25	24	ord 392	. . . . . . . . . . 11 |- ( ( ( R1 ` A ) e. Tarski /\ A e. On ) -> ( -. A ~~ ( R1 ` A ) -> A e. ( R1 ` A ) ) )
26	20, 25	mt3d 140	. . . . . . . . . 10 |- ( ( ( R1 ` A ) e. Tarski /\ A e. On ) -> A ~~ ( R1 ` A ) )
27	2, 3, 26	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> A ~~ ( R1 ` A ) )
28		carden2b 8793	. . . . . . . . 9 |- ( A ~~ ( R1 ` A ) -> ( card ` A ) = ( card ` ( R1 ` A ) ) )
29	27, 28	syl 17	. . . . . . . 8 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( card ` A ) = ( card ` ( R1 ` A ) ) )
30		simpl 473	. . . . . . . . . 10 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> A e. On )
31		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> ( R1 ` A ) e. Tarski )
32	22	adantr 481	. . . . . . . . . . . . . 14 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> A C_ ( R1 ` A ) )
33	32	sselda 3603	. . . . . . . . . . . . 13 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> x e. ( R1 ` A ) )
34		tsksdom 9578	. . . . . . . . . . . . 13 |- ( ( ( R1 ` A ) e. Tarski /\ x e. ( R1 ` A ) ) -> x ~< ( R1 ` A ) )
35	31, 33, 34	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> x ~< ( R1 ` A ) )
36		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> A e. On )
37	26	ensymd 8007	. . . . . . . . . . . . 13 |- ( ( ( R1 ` A ) e. Tarski /\ A e. On ) -> ( R1 ` A ) ~~ A )
38	31, 36, 37	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> ( R1 ` A ) ~~ A )
39		sdomentr 8094	. . . . . . . . . . . 12 |- ( ( x ~< ( R1 ` A ) /\ ( R1 ` A ) ~~ A ) -> x ~< A )
40	35, 38, 39	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ x e. A ) -> x ~< A )
41	40	ralrimiva 2966	. . . . . . . . . 10 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> A. x e. A x ~< A )
42		iscard 8801	. . . . . . . . . 10 |- ( ( card ` A ) = A <-> ( A e. On /\ A. x e. A x ~< A ) )
43	30, 41, 42	sylanbrc 698	. . . . . . . . 9 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> ( card ` A ) = A )
44	43	adantr 481	. . . . . . . 8 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( card ` A ) = A )
45	29, 44	eqtr3d 2658	. . . . . . 7 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( card ` ( R1 ` A ) ) = A )
46		r10 8631	. . . . . . . . . . 11 |- ( R1 ` (/) ) = (/)
47		on0eln0 5780	. . . . . . . . . . . . 13 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
48	47	biimpar 502	. . . . . . . . . . . 12 |- ( ( A e. On /\ A =/= (/) ) -> (/) e. A )
49		r1sdom 8637	. . . . . . . . . . . 12 |- ( ( A e. On /\ (/) e. A ) -> ( R1 ` (/) ) ~< ( R1 ` A ) )
50	48, 49	syldan 487	. . . . . . . . . . 11 |- ( ( A e. On /\ A =/= (/) ) -> ( R1 ` (/) ) ~< ( R1 ` A ) )
51	46, 50	syl5eqbrr 4689	. . . . . . . . . 10 |- ( ( A e. On /\ A =/= (/) ) -> (/) ~< ( R1 ` A ) )
52		fvex 6201	. . . . . . . . . . 11 |- ( R1 ` A ) e. _V
53	52	0sdom 8091	. . . . . . . . . 10 |- ( (/) ~< ( R1 ` A ) <-> ( R1 ` A ) =/= (/) )
54	51, 53	sylib 208	. . . . . . . . 9 |- ( ( A e. On /\ A =/= (/) ) -> ( R1 ` A ) =/= (/) )
55	54	adantlr 751	. . . . . . . 8 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( R1 ` A ) =/= (/) )
56		tskcard 9603	. . . . . . . 8 |- ( ( ( R1 ` A ) e. Tarski /\ ( R1 ` A ) =/= (/) ) -> ( card ` ( R1 ` A ) ) e. Inacc )
57	2, 55, 56	syl2anc 693	. . . . . . 7 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> ( card ` ( R1 ` A ) ) e. Inacc )
58	45, 57	eqeltrrd 2702	. . . . . 6 |- ( ( ( A e. On /\ ( R1 ` A ) e. Tarski ) /\ A =/= (/) ) -> A e. Inacc )
59	58	ex 450	. . . . 5 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> ( A =/= (/) -> A e. Inacc ) )
60	1, 59	syl5bir 233	. . . 4 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> ( -. A = (/) -> A e. Inacc ) )
61	60	orrd 393	. . 3 |- ( ( A e. On /\ ( R1 ` A ) e. Tarski ) -> ( A = (/) \/ A e. Inacc ) )
62	61	ex 450	. 2 |- ( A e. On -> ( ( R1 ` A ) e. Tarski -> ( A = (/) \/ A e. Inacc ) ) )
63		fveq2 6191	. . . . 5 |- ( A = (/) -> ( R1 ` A ) = ( R1 ` (/) ) )
64	63, 46	syl6eq 2672	. . . 4 |- ( A = (/) -> ( R1 ` A ) = (/) )
65		0tsk 9577	. . . 4 |- (/) e. Tarski
66	64, 65	syl6eqel 2709	. . 3 |- ( A = (/) -> ( R1 ` A ) e. Tarski )
67		inatsk 9600	. . 3 |- ( A e. Inacc -> ( R1 ` A ) e. Tarski )
68	66, 67	jaoi 394	. 2 |- ( ( A = (/) \/ A e. Inacc ) -> ( R1 ` A ) e. Tarski )
69	62, 68	impbid1 215	1 |- ( A e. On -> ( ( R1 ` A ) e. Tarski <-> ( A = (/) \/ A e. Inacc ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   dom cdm 5114   "cima 5117   Oncon0 5723   suc csuc 5725   Fn wfn 5883   ` cfv 5888   ~~ cen 7952   ~< csdm 7954   R1cr1 8625   rankcrnk 8626   cardccrd 8761   Inacccina 9505   Tarskictsk 9570

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  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iin 4523  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-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-r1 8627  df-rank 8628  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768  df-ac 8939  df-wina 9506  df-ina 9507  df-tsk 9571

This theorem is referenced by: (None)