Theorem cfsuc 5065

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 3174	. . 3 |- (A e. On <-> suc A e. On)
2		cfval 5056	. . 3 |- (suc A e. On -> (cf` suc A) = |^|{x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
3	1, 2	sylbi 197	. 2 |- (A e. On -> (cf` suc A) = |^|{x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
4		snex 2826	. . . . . . 7 |- {A} e. V
5		fveq2 3835	. . . . . . . . 9 |- (y = {A} -> (card` y) = (card` {A}))
6	5	eqeq2d 1529	. . . . . . . 8 |- (y = {A} -> (1o = (card` y) <-> 1o = (card` {A})))
7		sseq1 2134	. . . . . . . . 9 |- (y = {A} -> (y (_ suc A <-> {A} (_ suc A))
8		rexeq1 1833	. . . . . . . . . 10 |- (y = {A} -> (E.w e. y z (_ w <-> E.w e. {A}z (_ w))
9	8	ralbidv 1709	. . . . . . . . 9 |- (y = {A} -> (A.z e. suc AE.w e. y z (_ w <-> A.z e. suc AE.w e. {A}z (_ w))
10	7, 9	anbi12d 631	. . . . . . . 8 |- (y = {A} -> ((y (_ suc A /\ A.z e. suc AE.w e. y z (_ w) <-> ({A} (_ suc A /\ A.z e. suc AE.w e. {A}z (_ w)))
11	6, 10	anbi12d 631	. . . . . . 7 |- (y = {A} -> ((1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> (1o = (card` {A}) /\ ({A} (_ suc A /\ A.z e. suc AE.w e. {A}z (_ w))))
12	4, 11	cla4ev 1915	. . . . . 6 |- ((1o = (card` {A}) /\ ({A} (_ suc A /\ A.z e. suc AE.w e. {A}z (_ w)) -> E.y(1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)))
13		cardsn 4982	. . . . . . 7 |- (A e. On -> (card` {A}) = 1o)
14	13	eqcomd 1523	. . . . . 6 |- (A e. On -> 1o = (card` {A}))
15		onelss 3017	. . . . . . . . . . . 12 |- (A e. On -> (z e. A -> z (_ A))
16		eqimss 2161	. . . . . . . . . . . . 13 |- (z = A -> z (_ A)
17	16	a1i 8	. . . . . . . . . . . 12 |- (A e. On -> (z = A -> z (_ A))
18	15, 17	jaod 424	. . . . . . . . . . 11 |- (A e. On -> ((z e. A \/ z = A) -> z (_ A))
19		elsuci 3039	. . . . . . . . . . 11 |- (z e. suc A -> (z e. A \/ z = A))
20	18, 19	syl5 21	. . . . . . . . . 10 |- (A e. On -> (z e. suc A -> z (_ A))
21		snidg 2494	. . . . . . . . . 10 |- (A e. On -> A e. {A})
22	20, 21	jctild 604	. . . . . . . . 9 |- (A e. On -> (z e. suc A -> (A e. {A} /\ z (_ A)))
23		sseq2 2135	. . . . . . . . . 10 |- (w = A -> (z (_ w <-> z (_ A))
24	23	rcla4ev 1923	. . . . . . . . 9 |- ((A e. {A} /\ z (_ A) -> E.w e. {A}z (_ w)
25	22, 24	syl6 22	. . . . . . . 8 |- (A e. On -> (z e. suc A -> E.w e. {A}z (_ w))
26	25	r19.21aiv 1759	. . . . . . 7 |- (A e. On -> A.z e. suc AE.w e. {A}z (_ w)
27		ssun2 2246	. . . . . . . 8 |- {A} (_ (A u. {A})
28		df-suc 2981	. . . . . . . 8 |- suc A = (A u. {A})
29	27, 28	sseqtr4i 2146	. . . . . . 7 |- {A} (_ suc A
30	26, 29	jctil 290	. . . . . 6 |- (A e. On -> ({A} (_ suc A /\ A.z e. suc AE.w e. {A}z (_ w))
31	12, 14, 30	sylanc 473	. . . . 5 |- (A e. On -> E.y(1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)))
32		1on 4274	. . . . . . 7 |- 1o e. On
33	32	elisseti 1864	. . . . . 6 |- 1o e. V
34		eqeq1 1524	. . . . . . . 8 |- (x = 1o -> (x = (card` y) <-> 1o = (card` y)))
35	34	anbi1d 620	. . . . . . 7 |- (x = 1o -> ((x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> (1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))))
36	35	exbidv 1317	. . . . . 6 |- (x = 1o -> (E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> E.y(1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))))
37	33, 36	elab 1943	. . . . 5 |- (1o e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} <-> E.y(1o = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)))
38	31, 37	sylibr 198	. . . 4 |- (A e. On -> 1o e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
39		el1o 4282	. . . . . 6 |- (v e. 1o <-> v = (/))
40		eqcom 1520	. . . . . . . . . . . . 13 |- ((/) = (card` y) <-> (card` y) = (/))
41		visset 1859	. . . . . . . . . . . . . 14 |- y e. V
42		cardeq0 4980	. . . . . . . . . . . . . 14 |- (y e. V -> ((card` y) = (/) <-> y = (/)))
43	41, 42	ax-mp 7	. . . . . . . . . . . . 13 |- ((card` y) = (/) <-> y = (/))
44	40, 43	bitri 171	. . . . . . . . . . . 12 |- ((/) = (card` y) <-> y = (/))
45		rex0 2344	. . . . . . . . . . . . . . . 16 |- -. E.w e. (/) z (_ w
46	45	a1i 8	. . . . . . . . . . . . . . 15 |- (z e. suc A -> -. E.w e. (/) z (_ w)
47	46	nrex 1775	. . . . . . . . . . . . . 14 |- -. E.z e. suc AE.w e. (/) z (_ w
48		nsuceq0 3053	. . . . . . . . . . . . . . 15 |- suc A =/= (/)
49		r19.2z 2401	. . . . . . . . . . . . . . 15 |- ((suc A =/= (/) /\ A.z e. suc AE.w e. (/) z (_ w) -> E.z e. suc AE.w e. (/) z (_ w)
50	48, 49	mpan 699	. . . . . . . . . . . . . 14 |- (A.z e. suc AE.w e. (/) z (_ w -> E.z e. suc AE.w e. (/) z (_ w)
51	47, 50	mto 105	. . . . . . . . . . . . 13 |- -. A.z e. suc AE.w e. (/) z (_ w
52		rexeq1 1833	. . . . . . . . . . . . . 14 |- (y = (/) -> (E.w e. y z (_ w <-> E.w e. (/) z (_ w))
53	52	ralbidv 1709	. . . . . . . . . . . . 13 |- (y = (/) -> (A.z e. suc AE.w e. y z (_ w <-> A.z e. suc AE.w e. (/) z (_ w))
54	51, 53	mtbiri 722	. . . . . . . . . . . 12 |- (y = (/) -> -. A.z e. suc AE.w e. y z (_ w)
55	44, 54	sylbi 197	. . . . . . . . . . 11 |- ((/) = (card` y) -> -. A.z e. suc AE.w e. y z (_ w)
56	55	intnand 697	. . . . . . . . . 10 |- ((/) = (card` y) -> -. (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))
57		imnan 240	. . . . . . . . . 10 |- (((/) = (card` y) -> -. (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> -. ((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)))
58	56, 57	mpbi 187	. . . . . . . . 9 |- -. ((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))
59	58	nex 1137	. . . . . . . 8 |- -. E.y((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))
60		0ex 2785	. . . . . . . . 9 |- (/) e. V
61		eqeq1 1524	. . . . . . . . . . 11 |- (x = (/) -> (x = (card` y) <-> (/) = (card` y)))
62	61	anbi1d 620	. . . . . . . . . 10 |- (x = (/) -> ((x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> ((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))))
63	62	exbidv 1317	. . . . . . . . 9 |- (x = (/) -> (E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) <-> E.y((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))))
64	60, 63	elab 1943	. . . . . . . 8 |- ((/) e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} <-> E.y((/) = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)))
65	59, 64	mtbir 190	. . . . . . 7 |- -. (/) e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}
66		eleq1 1577	. . . . . . 7 |- (v = (/) -> (v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} <-> (/) e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}))
67	65, 66	mtbiri 722	. . . . . 6 |- (v = (/) -> -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
68	39, 67	sylbi 197	. . . . 5 |- (v e. 1o -> -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
69	68	rgen 1744	. . . 4 |- A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}
70	38, 69	jctir 291	. . 3 |- (A e. On -> (1o e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} /\ A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}))
71		cardon 4974	. . . . . . . 8 |- (card` y) e. On
72		eleq1 1577	. . . . . . . 8 |- (x = (card` y) -> (x e. On <-> (card` y) e. On))
73	71, 72	mpbiri 192	. . . . . . 7 |- (x = (card` y) -> x e. On)
74	73	adantr 389	. . . . . 6 |- ((x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) -> x e. On)
75	74	19.23aiv 1333	. . . . 5 |- (E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w)) -> x e. On)
76	75	abssi 2174	. . . 4 |- {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} (_ On
77		oneqmini 3024	. . . 4 |- ({x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} (_ On -> ((1o e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} /\ A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}) -> 1o = |^|{x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}))
78	76, 77	ax-mp 7	. . 3 |- ((1o e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))} /\ A.v e. 1o -. v e. {x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))}) -> 1o = |^|{x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
79	70, 78	syl 10	. 2 |- (A e. On -> 1o = |^|{x | E.y(x = (card` y) /\ (y (_ suc A /\ A.z e. suc AE.w e. y z (_ w))})
80	3, 79	eqtr4d 1553	1 |- (A e. On -> (cf` suc A) = 1o)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  E.wex 1016  {cab 1505   =/= wne 1628  A.wral 1691  E.wrex 1692  Vcvv 1857   u. cun 2097   (_ wss 2099  (/)c0 2332  {csn 2467  |^|cint 2600  Oncon0 2975  suc csuc 2977  ` cfv 3263  1oc1o 4264  cardccrd 4959  cfccf 4961

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-1o 4269  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-card 4962  df-cf 4964

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