Theorem cfeq0 6062

Description: Only the ordinal zero has cofinality zero.

Assertion

Ref	Expression
cfeq0	|- (A e. On -> ((cf` A) = (/) <-> A = (/)))

Proof of Theorem cfeq0

Step	Hyp	Ref	Expression
1		cfval 6054	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
2	1	eqeq1d 1892	. . 3 |- (A e. On -> ((cf` A) = (/) <-> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} = (/)))
3		visset 2295	. . . . . . . . 9 |- v e. _V
4		eqeq1 1890	. . . . . . . . . . 11 |- (x = v -> (x = (card` y) <-> v = (card` y)))
5	4	anbi1d 679	. . . . . . . . . 10 |- (x = v -> ((x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> (v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
6	5	exbidv 1657	. . . . . . . . 9 |- (x = v -> (E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
7	3, 6	elab 2403	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} <-> E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)))
8		fveq2 4681	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = (card` (card` y)))
9		cardidm 6001	. . . . . . . . . . . 12 |- (card` (card` y)) = (card` y)
10	8, 9	syl6eq 1944	. . . . . . . . . . 11 |- (v = (card` y) -> (card` v) = (card` y))
11		eqeq2 1893	. . . . . . . . . . 11 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
12	10, 11	mpbird 213	. . . . . . . . . 10 |- (v = (card` y) -> (card` v) = v)
13	12	adantr 425	. . . . . . . . 9 |- ((v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> (card` v) = v)
14	13	19.23aiv 1674	. . . . . . . 8 |- (E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> (card` v) = v)
15	7, 14	sylbi 216	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> (card` v) = v)
16		cardon 5976	. . . . . . 7 |- (card` v) e. On
17	15, 16	syl6eqelr 1980	. . . . . 6 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> v e. On)
18	17	ssriv 2621	. . . . 5 |- {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ On
19		onint0 3877	. . . . 5 |- ({x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ On -> (|^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} = (/) <-> (/) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))}))
20	18, 19	ax-mp 7	. . . 4 |- (|^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} = (/) <-> (/) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
21		0ex 3446	. . . . . 6 |- (/) e. _V
22		eqeq1 1890	. . . . . . . 8 |- (x = (/) -> (x = (card` y) <-> (/) = (card` y)))
23	22	anbi1d 679	. . . . . . 7 |- (x = (/) -> ((x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> ((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
24	23	exbidv 1657	. . . . . 6 |- (x = (/) -> (E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> E.y((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
25	21, 24	elab 2403	. . . . 5 |- ((/) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} <-> E.y((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)))
26		sseq1 2637	. . . . . . . . . 10 |- (y = (/) -> (y C_ A <-> (/) C_ A))
27		rexeq 2267	. . . . . . . . . . 11 |- (y = (/) -> (E.w e. y z C_ w <-> E.w e. (/) z C_ w))
28	27	ralbidv 2123	. . . . . . . . . 10 |- (y = (/) -> (A.z e. A E.w e. y z C_ w <-> A.z e. A E.w e. (/) z C_ w))
29	26, 28	anbi12d 690	. . . . . . . . 9 |- (y = (/) -> ((y C_ A /\ A.z e. A E.w e. y z C_ w) <-> ((/) C_ A /\ A.z e. A E.w e. (/) z C_ w)))
30	29	biimpa 460	. . . . . . . 8 |- ((y = (/) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> ((/) C_ A /\ A.z e. A E.w e. (/) z C_ w))
31		eqcom 1886	. . . . . . . . 9 |- ((/) = (card` y) <-> (card` y) = (/))
32		visset 2295	. . . . . . . . . 10 |- y e. _V
33		cardeq0 5982	. . . . . . . . . 10 |- (y e. _V -> ((card` y) = (/) <-> y = (/)))
34	32, 33	ax-mp 7	. . . . . . . . 9 |- ((card` y) = (/) <-> y = (/))
35	31, 34	bitri 190	. . . . . . . 8 |- ((/) = (card` y) <-> y = (/))
36	30, 35	sylanb 498	. . . . . . 7 |- (((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> ((/) C_ A /\ A.z e. A E.w e. (/) z C_ w))
37		rex0 2888	. . . . . . . . . . . . 13 |- -. E.w e. (/) z C_ w
38	37	a1i 8	. . . . . . . . . . . 12 |- (z e. A -> -. E.w e. (/) z C_ w)
39	38	rgen 2159	. . . . . . . . . . 11 |- A.z e. A -. E.w e. (/) z C_ w
40		r19.2z 2958	. . . . . . . . . . 11 |- ((A =/= (/) /\ A.z e. A -. E.w e. (/) z C_ w) -> E.z e. A -. E.w e. (/) z C_ w)
41	39, 40	mpan2 760	. . . . . . . . . 10 |- (A =/= (/) -> E.z e. A -. E.w e. (/) z C_ w)
42		rexnal 2114	. . . . . . . . . 10 |- (E.z e. A -. E.w e. (/) z C_ w <-> -. A.z e. A E.w e. (/) z C_ w)
43	41, 42	sylib 215	. . . . . . . . 9 |- (A =/= (/) -> -. A.z e. A E.w e. (/) z C_ w)
44	43	necon4ai 2067	. . . . . . . 8 |- (A.z e. A E.w e. (/) z C_ w -> A = (/))
45	44	adantl 424	. . . . . . 7 |- (((/) C_ A /\ A.z e. A E.w e. (/) z C_ w) -> A = (/))
46	36, 45	syl 12	. . . . . 6 |- (((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> A = (/))
47	46	19.23aiv 1674	. . . . 5 |- (E.y((/) = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> A = (/))
48	25, 47	sylbi 216	. . . 4 |- ((/) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> A = (/))
49	20, 48	sylbi 216	. . 3 |- (|^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} = (/) -> A = (/))
50	2, 49	syl6bi 231	. 2 |- (A e. On -> ((cf` A) = (/) -> A = (/)))
51		fveq2 4681	. . 3 |- (A = (/) -> (cf` A) = (cf` (/)))
52		cf0 6058	. . 3 |- (cf` (/)) = (/)
53	51, 52	syl6eq 1944	. 2 |- (A = (/) -> (cf` A) = (/))
54	50, 53	impbid1 575	1 |- (A e. On -> ((cf` A) = (/) <-> A = (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  |^|cint 3214  Oncon0 3657  ` cfv 3998  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-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-suc 3663  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-er 5318  df-en 5427  df-card 5862  df-cf 5864

Copyright terms: Public domain