Theorem cfsuc 6063

Description: Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cfsuc	|- (A e. On -> (cf` suc A) = 1o)

Proof of Theorem cfsuc

Step	Hyp	Ref	Expression
1		sucelon 3898	. . 3 |- (A e. On <-> suc A e. On)
2		cfval 6054	. . 3 |- (suc A e. On -> (cf` suc A) = |^|{x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
3	1, 2	sylbi 216	. 2 |- (A e. On -> (cf` suc A) = |^|{x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
4		cardon 5976	. . . . . . . 8 |- (card` y) e. On
5		eleq1 1957	. . . . . . . 8 |- (x = (card` y) -> (x e. On <-> (card` y) e. On))
6	4, 5	mpbiri 211	. . . . . . 7 |- (x = (card` y) -> x e. On)
7	6	adantr 425	. . . . . 6 |- ((x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) -> x e. On)
8	7	19.23aiv 1674	. . . . 5 |- (E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) -> x e. On)
9	8	abssi 2682	. . . 4 |- {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} C_ On
10		oneqmini 3714	. . . 4 |- ({x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} C_ On -> ((1o e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} /\ A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}) -> 1o = |^|{x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}))
11	9, 10	ax-mp 7	. . 3 |- ((1o e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} /\ A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}) -> 1o = |^|{x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
12		cardsn 5984	. . . . . 6 |- (A e. On -> (card` {A}) = 1o)
13	12	eqcomd 1889	. . . . 5 |- (A e. On -> 1o = (card` {A}))
14		onelss 3705	. . . . . . . . . . 11 |- (A e. On -> (z e. A -> z C_ A))
15		eqimss 2665	. . . . . . . . . . . 12 |- (z = A -> z C_ A)
16	15	a1i 8	. . . . . . . . . . 11 |- (A e. On -> (z = A -> z C_ A))
17	14, 16	jaod 469	. . . . . . . . . 10 |- (A e. On -> ((z e. A \/ z = A) -> z C_ A))
18		elsuci 3731	. . . . . . . . . 10 |- (z e. suc A -> (z e. A \/ z = A))
19	17, 18	syl5 20	. . . . . . . . 9 |- (A e. On -> (z e. suc A -> z C_ A))
20		snidg 3067	. . . . . . . . 9 |- (A e. On -> A e. {A})
21	19, 20	jctild 662	. . . . . . . 8 |- (A e. On -> (z e. suc A -> (A e. {A} /\ z C_ A)))
22		sseq2 2639	. . . . . . . . 9 |- (w = A -> (z C_ w <-> z C_ A))
23	22	rcla4ev 2381	. . . . . . . 8 |- ((A e. {A} /\ z C_ A) -> E.w e. {A}z C_ w)
24	21, 23	syl6 25	. . . . . . 7 |- (A e. On -> (z e. suc A -> E.w e. {A}z C_ w))
25	24	r19.21aiv 2175	. . . . . 6 |- (A e. On -> A.z e. suc AE.w e. {A}z C_ w)
26		ssun2 2768	. . . . . . 7 |- {A} C_ (A u. {A})
27		df-suc 3663	. . . . . . 7 |- suc A = (A u. {A})
28	26, 27	sseqtr4i 2650	. . . . . 6 |- {A} C_ suc A
29	25, 28	jctil 316	. . . . 5 |- (A e. On -> ({A} C_ suc A /\ A.z e. suc AE.w e. {A}z C_ w))
30		snex 3492	. . . . . 6 |- {A} e. _V
31		fveq2 4681	. . . . . . . 8 |- (y = {A} -> (card` y) = (card` {A}))
32	31	eqeq2d 1895	. . . . . . 7 |- (y = {A} -> (1o = (card` y) <-> 1o = (card` {A})))
33		sseq1 2637	. . . . . . . 8 |- (y = {A} -> (y C_ suc A <-> {A} C_ suc A))
34		rexeq 2267	. . . . . . . . 9 |- (y = {A} -> (E.w e. y z C_ w <-> E.w e. {A}z C_ w))
35	34	ralbidv 2123	. . . . . . . 8 |- (y = {A} -> (A.z e. suc AE.w e. y z C_ w <-> A.z e. suc AE.w e. {A}z C_ w))
36	33, 35	anbi12d 690	. . . . . . 7 |- (y = {A} -> ((y C_ suc A /\ A.z e. suc AE.w e. y z C_ w) <-> ({A} C_ suc A /\ A.z e. suc AE.w e. {A}z C_ w)))
37	32, 36	anbi12d 690	. . . . . 6 |- (y = {A} -> ((1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> (1o = (card` {A}) /\ ({A} C_ suc A /\ A.z e. suc AE.w e. {A}z C_ w))))
38	30, 37	cla4ev 2371	. . . . 5 |- ((1o = (card` {A}) /\ ({A} C_ suc A /\ A.z e. suc AE.w e. {A}z C_ w)) -> E.y(1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)))
39	13, 29, 38	syl11anc 524	. . . 4 |- (A e. On -> E.y(1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)))
40		1on 5182	. . . . . 6 |- 1o e. On
41	40	elisseti 2301	. . . . 5 |- 1o e. _V
42		eqeq1 1890	. . . . . . 7 |- (x = 1o -> (x = (card` y) <-> 1o = (card` y)))
43	42	anbi1d 679	. . . . . 6 |- (x = 1o -> ((x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> (1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))))
44	43	exbidv 1657	. . . . 5 |- (x = 1o -> (E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> E.y(1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))))
45	41, 44	elab 2403	. . . 4 |- (1o e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} <-> E.y(1o = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)))
46	39, 45	sylibr 217	. . 3 |- (A e. On -> 1o e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
47		el1o 5191	. . . . 5 |- (v e. 1o <-> v = (/))
48		eqcom 1886	. . . . . . . . . . . 12 |- ((/) = (card` y) <-> (card` y) = (/))
49		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
50		cardeq0 5982	. . . . . . . . . . . . 13 |- (y e. _V -> ((card` y) = (/) <-> y = (/)))
51	49, 50	ax-mp 7	. . . . . . . . . . . 12 |- ((card` y) = (/) <-> y = (/))
52	48, 51	bitri 190	. . . . . . . . . . 11 |- ((/) = (card` y) <-> y = (/))
53		rex0 2888	. . . . . . . . . . . . . . 15 |- -. E.w e. (/) z C_ w
54	53	a1i 8	. . . . . . . . . . . . . 14 |- (z e. suc A -> -. E.w e. (/) z C_ w)
55	54	nrex 2192	. . . . . . . . . . . . 13 |- -. E.z e. suc AE.w e. (/) z C_ w
56		nsuceq0 3749	. . . . . . . . . . . . . 14 |- suc A =/= (/)
57		r19.2z 2958	. . . . . . . . . . . . . 14 |- ((suc A =/= (/) /\ A.z e. suc AE.w e. (/) z C_ w) -> E.z e. suc AE.w e. (/) z C_ w)
58	56, 57	mpan 759	. . . . . . . . . . . . 13 |- (A.z e. suc AE.w e. (/) z C_ w -> E.z e. suc AE.w e. (/) z C_ w)
59	55, 58	mto 121	. . . . . . . . . . . 12 |- -. A.z e. suc AE.w e. (/) z C_ w
60		rexeq 2267	. . . . . . . . . . . . 13 |- (y = (/) -> (E.w e. y z C_ w <-> E.w e. (/) z C_ w))
61	60	ralbidv 2123	. . . . . . . . . . . 12 |- (y = (/) -> (A.z e. suc AE.w e. y z C_ w <-> A.z e. suc AE.w e. (/) z C_ w))
62	59, 61	mtbiri 785	. . . . . . . . . . 11 |- (y = (/) -> -. A.z e. suc AE.w e. y z C_ w)
63	52, 62	sylbi 216	. . . . . . . . . 10 |- ((/) = (card` y) -> -. A.z e. suc AE.w e. y z C_ w)
64	63	intnand 757	. . . . . . . . 9 |- ((/) = (card` y) -> -. (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))
65		imnan 261	. . . . . . . . 9 |- (((/) = (card` y) -> -. (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> -. ((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)))
66	64, 65	mpbi 206	. . . . . . . 8 |- -. ((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))
67	66	nex 1456	. . . . . . 7 |- -. E.y((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))
68		0ex 3446	. . . . . . . 8 |- (/) e. _V
69		eqeq1 1890	. . . . . . . . . 10 |- (x = (/) -> (x = (card` y) <-> (/) = (card` y)))
70	69	anbi1d 679	. . . . . . . . 9 |- (x = (/) -> ((x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> ((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))))
71	70	exbidv 1657	. . . . . . . 8 |- (x = (/) -> (E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)) <-> E.y((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))))
72	68, 71	elab 2403	. . . . . . 7 |- ((/) e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} <-> E.y((/) = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w)))
73	67, 72	mtbir 209	. . . . . 6 |- -. (/) e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}
74		eleq1 1957	. . . . . 6 |- (v = (/) -> (v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))} <-> (/) e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}))
75	73, 74	mtbiri 785	. . . . 5 |- (v = (/) -> -. v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
76	47, 75	sylbi 216	. . . 4 |- (v e. 1o -> -. v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
77	76	rgen 2159	. . 3 |- A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))}
78	11, 46, 77	sylancl 525	. 2 |- (A e. On -> 1o = |^|{x | E.y(x = (card` y) /\ (y C_ suc A /\ A.z e. suc AE.w e. y z C_ w))})
79	3, 78	eqtr4d 1928	1 |- (A e. On -> (cf` suc A) = 1o)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044  |^|cint 3214  Oncon0 3657  suc csuc 3659  ` cfv 3998  1oc1o 5172  cardccrd 5859  cfccf 5861

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-ac 5906

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-reu 2111  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862  df-cf 5864

Copyright terms: Public domain