Theorem rankr1 5785

Description: A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79.

Hypothesis

Ref	Expression
rankr1.1	|- A e. _V

Assertion

Ref	Expression
rankr1	|- (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B)))

Proof of Theorem rankr1

Step	Hyp	Ref	Expression
1		rankon 5782	. . . . 5 |- (rank` A) e. On
2		eleq1 1957	. . . . 5 |- (B = (rank` A) -> (B e. On <-> (rank` A) e. On))
3	1, 2	mpbiri 211	. . . 4 |- (B = (rank` A) -> B e. On)
4		eloni 3667	. . . . 5 |- (B e. On -> Ord B)
5		ordzsl 3927	. . . . . 6 |- (Ord B <-> (B = (/) \/ E.x e. On B = suc x \/ Lim B))
6		noel 2879	. . . . . . . . 9 |- -. A e. (/)
7		fveq2 4681	. . . . . . . . . . 11 |- (B = (/) -> (R1` B) = (R1` (/)))
8		r10 5762	. . . . . . . . . . 11 |- (R1` (/)) = (/)
9	7, 8	syl6eq 1944	. . . . . . . . . 10 |- (B = (/) -> (R1` B) = (/))
10	9	eleq2d 1964	. . . . . . . . 9 |- (B = (/) -> (A e. (R1` B) <-> A e. (/)))
11	6, 10	mtbiri 785	. . . . . . . 8 |- (B = (/) -> -. A e. (R1` B))
12	11	a1d 15	. . . . . . 7 |- (B = (/) -> (B = (rank` A) -> -. A e. (R1` B)))
13		visset 2295	. . . . . . . . . . . . . . 15 |- x e. _V
14	13	sucid 3744	. . . . . . . . . . . . . 14 |- x e. suc x
15		eleq2 1958	. . . . . . . . . . . . . 14 |- (B = suc x -> (x e. B <-> x e. suc x))
16	14, 15	mpbiri 211	. . . . . . . . . . . . 13 |- (B = suc x -> x e. B)
17	16	adantl 424	. . . . . . . . . . . 12 |- ((x e. On /\ B = suc x) -> x e. B)
18		eleq1a 1966	. . . . . . . . . . . . . . 15 |- (suc x e. On -> (B = suc x -> B e. On))
19	18	imp 377	. . . . . . . . . . . . . 14 |- ((suc x e. On /\ B = suc x) -> B e. On)
20		suceloni 3894	. . . . . . . . . . . . . 14 |- (x e. On -> suc x e. On)
21	19, 20	sylan 497	. . . . . . . . . . . . 13 |- ((x e. On /\ B = suc x) -> B e. On)
22		simpl 346	. . . . . . . . . . . . 13 |- ((x e. On /\ B = suc x) -> x e. On)
23		ontri1 3695	. . . . . . . . . . . . . 14 |- ((B e. On /\ x e. On) -> (B C_ x <-> -. x e. B))
24	23	con2bid 585	. . . . . . . . . . . . 13 |- ((B e. On /\ x e. On) -> (x e. B <-> -. B C_ x))
25	21, 22, 24	syl11anc 524	. . . . . . . . . . . 12 |- ((x e. On /\ B = suc x) -> (x e. B <-> -. B C_ x))
26	17, 25	mpbid 212	. . . . . . . . . . 11 |- ((x e. On /\ B = suc x) -> -. B C_ x)
27	26	adantr 425	. . . . . . . . . 10 |- (((x e. On /\ B = suc x) /\ B = (rank` A)) -> -. B C_ x)
28		rankr1.1	. . . . . . . . . . . . . . . . . . 19 |- A e. _V
29	28	rankval 5779	. . . . . . . . . . . . . . . . . 18 |- (rank` A) = |^|{y e. On | A e. (R1` suc y)}
30	29	eqeq2i 1894	. . . . . . . . . . . . . . . . 17 |- (B = (rank` A) <-> B = |^|{y e. On | A e. (R1` suc y)})
31	30	biimpi 168	. . . . . . . . . . . . . . . 16 |- (B = (rank` A) -> B = |^|{y e. On | A e. (R1` suc y)})
32	31	sseq1d 2644	. . . . . . . . . . . . . . 15 |- (B = (rank` A) -> (B C_ x <-> |^|{y e. On | A e. (R1` suc y)} C_ x))
33		suceq 3729	. . . . . . . . . . . . . . . . . . 19 |- (y = x -> suc y = suc x)
34	33	fveq2d 4685	. . . . . . . . . . . . . . . . . 18 |- (y = x -> (R1` suc y) = (R1` suc x))
35	34	eleq2d 1964	. . . . . . . . . . . . . . . . 17 |- (y = x -> (A e. (R1` suc y) <-> A e. (R1` suc x)))
36	35	onintss 3713	. . . . . . . . . . . . . . . 16 |- (x e. On -> (A e. (R1` suc x) -> |^|{y e. On | A e. (R1` suc y)} C_ x))
37	36	imp 377	. . . . . . . . . . . . . . 15 |- ((x e. On /\ A e. (R1` suc x)) -> |^|{y e. On | A e. (R1` suc y)} C_ x)
38	32, 37	syl5bir 227	. . . . . . . . . . . . . 14 |- (B = (rank` A) -> ((x e. On /\ A e. (R1` suc x)) -> B C_ x))
39		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (B = suc x -> (R1` B) = (R1` suc x))
40	39	eleq2d 1964	. . . . . . . . . . . . . . 15 |- (B = suc x -> (A e. (R1` B) <-> A e. (R1` suc x)))
41	40	biimpa 460	. . . . . . . . . . . . . 14 |- ((B = suc x /\ A e. (R1` B)) -> A e. (R1` suc x))
42	38, 41	sylan2i 514	. . . . . . . . . . . . 13 |- (B = (rank` A) -> ((x e. On /\ (B = suc x /\ A e. (R1` B))) -> B C_ x))
43	42	exp4d 412	. . . . . . . . . . . 12 |- (B = (rank` A) -> (x e. On -> (B = suc x -> (A e. (R1` B) -> B C_ x))))
44	43	com3l 38	. . . . . . . . . . 11 |- (x e. On -> (B = suc x -> (B = (rank` A) -> (A e. (R1` B) -> B C_ x))))
45	44	imp31 389	. . . . . . . . . 10 |- (((x e. On /\ B = suc x) /\ B = (rank` A)) -> (A e. (R1` B) -> B C_ x))
46	27, 45	mtod 123	. . . . . . . . 9 |- (((x e. On /\ B = suc x) /\ B = (rank` A)) -> -. A e. (R1` B))
47	46	exp31 407	. . . . . . . 8 |- (x e. On -> (B = suc x -> (B = (rank` A) -> -. A e. (R1` B))))
48	47	r19.23aiv 2211	. . . . . . 7 |- (E.x e. On B = suc x -> (B = (rank` A) -> -. A e. (R1` B)))
49		eleq2 1958	. . . . . . . . . . . . . . . . . 18 |- (B = (rank` A) -> (x e. B <-> x e. (rank` A)))
50	1	oneli 3777	. . . . . . . . . . . . . . . . . 18 |- (x e. (rank` A) -> x e. On)
51	49, 50	syl6bi 231	. . . . . . . . . . . . . . . . 17 |- (B = (rank` A) -> (x e. B -> x e. On))
52	51	anc2li 326	. . . . . . . . . . . . . . . 16 |- (B = (rank` A) -> (x e. B -> (B = (rank` A) /\ x e. On)))
53		r1ord2 5767	. . . . . . . . . . . . . . . . . . . . . 22 |- (suc x e. On -> (x e. suc x -> (R1` x) C_ (R1` suc x)))
54	14, 53	mpi 55	. . . . . . . . . . . . . . . . . . . . 21 |- (suc x e. On -> (R1` x) C_ (R1` suc x))
55	20, 54	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (x e. On -> (R1` x) C_ (R1` suc x))
56	55	sseld 2619	. . . . . . . . . . . . . . . . . . 19 |- (x e. On -> (A e. (R1` x) -> A e. (R1` suc x)))
57	29	sseq1i 2641	. . . . . . . . . . . . . . . . . . . 20 |- ((rank` A) C_ x <-> |^|{y e. On | A e. (R1` suc y)} C_ x)
58	36, 57	syl6ibr 230	. . . . . . . . . . . . . . . . . . 19 |- (x e. On -> (A e. (R1` suc x) -> (rank` A) C_ x))
59	56, 58	syld 30	. . . . . . . . . . . . . . . . . 18 |- (x e. On -> (A e. (R1` x) -> (rank` A) C_ x))
60		sseq1 2637	. . . . . . . . . . . . . . . . . . 19 |- (B = (rank` A) -> (B C_ x <-> (rank` A) C_ x))
61	60	biimprd 171	. . . . . . . . . . . . . . . . . 18 |- (B = (rank` A) -> ((rank` A) C_ x -> B C_ x))
62	59, 61	sylan9r 519	. . . . . . . . . . . . . . . . 17 |- ((B = (rank` A) /\ x e. On) -> (A e. (R1` x) -> B C_ x))
63	23, 3	sylan 497	. . . . . . . . . . . . . . . . 17 |- ((B = (rank` A) /\ x e. On) -> (B C_ x <-> -. x e. B))
64	62, 63	sylibd 219	. . . . . . . . . . . . . . . 16 |- ((B = (rank` A) /\ x e. On) -> (A e. (R1` x) -> -. x e. B))
65	52, 64	syl6 25	. . . . . . . . . . . . . . 15 |- (B = (rank` A) -> (x e. B -> (A e. (R1` x) -> -. x e. B)))
66	65	com3l 38	. . . . . . . . . . . . . 14 |- (x e. B -> (A e. (R1` x) -> (B = (rank` A) -> -. x e. B)))
67		con2 106	. . . . . . . . . . . . . 14 |- ((B = (rank` A) -> -. x e. B) -> (x e. B -> -. B = (rank` A)))
68	66, 67	syl6 25	. . . . . . . . . . . . 13 |- (x e. B -> (A e. (R1` x) -> (x e. B -> -. B = (rank` A))))
69	68	pm2.43a 80	. . . . . . . . . . . 12 |- (x e. B -> (A e. (R1` x) -> -. B = (rank` A)))
70	69	r19.23aiv 2211	. . . . . . . . . . 11 |- (E.x e. B A e. (R1` x) -> -. B = (rank` A))
71	70	con2i 113	. . . . . . . . . 10 |- (B = (rank` A) -> -. E.x e. B A e. (R1` x))
72	71	adantr 425	. . . . . . . . 9 |- ((B = (rank` A) /\ Lim B) -> -. E.x e. B A e. (R1` x))
73		r1lim 5764	. . . . . . . . . . . 12 |- ((B e. _V /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
74		fvex 4689	. . . . . . . . . . . . 13 |- (rank` A) e. _V
75		eleq1 1957	. . . . . . . . . . . . 13 |- (B = (rank` A) -> (B e. _V <-> (rank` A) e. _V))
76	74, 75	mpbiri 211	. . . . . . . . . . . 12 |- (B = (rank` A) -> B e. _V)
77	73, 76	sylan 497	. . . . . . . . . . 11 |- ((B = (rank` A) /\ Lim B) -> (R1` B) = U_x e. B (R1` x))
78	77	eleq2d 1964	. . . . . . . . . 10 |- ((B = (rank` A) /\ Lim B) -> (A e. (R1` B) <-> A e. U_x e. B (R1` x)))
79		eliun 3259	. . . . . . . . . 10 |- (A e. U_x e. B (R1` x) <-> E.x e. B A e. (R1` x))
80	78, 79	syl6bb 595	. . . . . . . . 9 |- ((B = (rank` A) /\ Lim B) -> (A e. (R1` B) <-> E.x e. B A e. (R1` x)))
81	72, 80	mtbird 783	. . . . . . . 8 |- ((B = (rank` A) /\ Lim B) -> -. A e. (R1` B))
82	81	expcom 403	. . . . . . 7 |- (Lim B -> (B = (rank` A) -> -. A e. (R1` B)))
83	12, 48, 82	3jaoi 1160	. . . . . 6 |- ((B = (/) \/ E.x e. On B = suc x \/ Lim B) -> (B = (rank` A) -> -. A e. (R1` B)))
84	5, 83	sylbi 216	. . . . 5 |- (Ord B -> (B = (rank` A) -> -. A e. (R1` B)))
85	4, 84	syl 12	. . . 4 |- (B e. On -> (B = (rank` A) -> -. A e. (R1` B)))
86	3, 85	mpcom 60	. . 3 |- (B = (rank` A) -> -. A e. (R1` B))
87		ssrab2 2692	. . . . . . 7 |- {y e. On | A e. (R1` suc y)} C_ On
88		rankwflem 5776	. . . . . . . . 9 |- (A e. _V -> E.y e. On A e. (R1` suc y))
89	28, 88	ax-mp 7	. . . . . . . 8 |- E.y e. On A e. (R1` suc y)
90		rabn0 2893	. . . . . . . 8 |- ({y e. On | A e. (R1` suc y)} =/= (/) <-> E.y e. On A e. (R1` suc y))
91	89, 90	mpbir 207	. . . . . . 7 |- {y e. On | A e. (R1` suc y)} =/= (/)
92		onint 3876	. . . . . . 7 |- (({y e. On | A e. (R1` suc y)} C_ On /\ {y e. On | A e. (R1` suc y)} =/= (/)) -> |^|{y e. On | A e. (R1` suc y)} e. {y e. On | A e. (R1` suc y)})
93	87, 91, 92	mp2an 761	. . . . . 6 |- |^|{y e. On | A e. (R1` suc y)} e. {y e. On | A e. (R1` suc y)}
94	29, 93	eqeltri 1967	. . . . 5 |- (rank` A) e. {y e. On | A e. (R1` suc y)}
95		eleq1 1957	. . . . 5 |- (B = (rank` A) -> (B e. {y e. On | A e. (R1` suc y)} <-> (rank` A) e. {y e. On | A e. (R1` suc y)}))
96	94, 95	mpbiri 211	. . . 4 |- (B = (rank` A) -> B e. {y e. On | A e. (R1` suc y)})
97		suceq 3729	. . . . . . . 8 |- (y = B -> suc y = suc B)
98	97	fveq2d 4685	. . . . . . 7 |- (y = B -> (R1` suc y) = (R1` suc B))
99	98	eleq2d 1964	. . . . . 6 |- (y = B -> (A e. (R1` suc y) <-> A e. (R1` suc B)))
100	99	elrab 2414	. . . . 5 |- (B e. {y e. On | A e. (R1` suc y)} <-> (B e. On /\ A e. (R1` suc B)))
101	100	simprbi 353	. . . 4 |- (B e. {y e. On | A e. (R1` suc y)} -> A e. (R1` suc B))
102	96, 101	syl 12	. . 3 |- (B = (rank` A) -> A e. (R1` suc B))
103	86, 102	jca 310	. 2 |- (B = (rank` A) -> (-. A e. (R1` B) /\ A e. (R1` suc B)))
104	28	rankr1lem 5784	. . . . . 6 |- (B e. On -> (-. A e. (R1` B) -> B C_ (rank` A)))
105	104	com12 14	. . . . 5 |- (-. A e. (R1` B) -> (B e. On -> B C_ (rank` A)))
106		elfvdm 4704	. . . . . 6 |- (A e. (R1` suc B) -> suc B e. dom R1)
107		r1fnon 5761	. . . . . . . . 9 |- R1 Fn On
108		fndm 4512	. . . . . . . . 9 |- (R1 Fn On -> dom R1 = On)
109	107, 108	ax-mp 7	. . . . . . . 8 |- dom R1 = On
110	109	eleq2i 1961	. . . . . . 7 |- (suc B e. dom R1 <-> suc B e. On)
111		sucelon 3898	. . . . . . 7 |- (B e. On <-> suc B e. On)
112	110, 111	bitr4i 193	. . . . . 6 |- (suc B e. dom R1 <-> B e. On)
113	106, 112	sylib 215	. . . . 5 |- (A e. (R1` suc B) -> B e. On)
114	105, 113	syl5 20	. . . 4 |- (-. A e. (R1` B) -> (A e. (R1` suc B) -> B C_ (rank` A)))
115	114	imp 377	. . 3 |- ((-. A e. (R1` B) /\ A e. (R1` suc B)) -> B C_ (rank` A))
116	99	onintss 3713	. . . . . 6 |- (B e. On -> (A e. (R1` suc B) -> |^|{y e. On | A e. (R1` suc y)} C_ B))
117	113, 116	mpcom 60	. . . . 5 |- (A e. (R1` suc B) -> |^|{y e. On | A e. (R1` suc y)} C_ B)
118	117, 29	syl5ss 2661	. . . 4 |- (A e. (R1` suc B) -> (rank` A) C_ B)
119	118	adantl 424	. . 3 |- ((-. A e. (R1` B) /\ A e. (R1` suc B)) -> (rank` A) C_ B)
120	115, 119	eqssd 2633	. 2 |- ((-. A e. (R1` B) /\ A e. (R1` suc B)) -> B = (rank` A))
121	103, 120	impbii 174	1 |- (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  |^|cint 3214  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  dom cdm 3986   Fn wfn 3993  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankr1g 5786  ssrankr1 5787  rankpw 5795

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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