Theorem cfub 6056

Description: An upper bound on cofinality.

Assertion

Ref	Expression
cfub	|- (cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))}

Distinct variable group:   x,y,A

Proof of Theorem cfub

Step	Hyp	Ref	Expression
1		cfval 6054	. . 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))})
2		ssel 2615	. . . . . . . . . . . . . . . . 17 |- (y C_ A -> (w e. y -> w e. A))
3		onelon 3683	. . . . . . . . . . . . . . . . . 18 |- ((A e. On /\ w e. A) -> w e. On)
4	3	ex 402	. . . . . . . . . . . . . . . . 17 |- (A e. On -> (w e. A -> w e. On))
5	2, 4	sylan9r 519	. . . . . . . . . . . . . . . 16 |- ((A e. On /\ y C_ A) -> (w e. y -> w e. On))
6		onelss 3705	. . . . . . . . . . . . . . . 16 |- (w e. On -> (z e. w -> z C_ w))
7	5, 6	syl6 25	. . . . . . . . . . . . . . 15 |- ((A e. On /\ y C_ A) -> (w e. y -> (z e. w -> z C_ w)))
8	7	imdistand 493	. . . . . . . . . . . . . 14 |- ((A e. On /\ y C_ A) -> ((w e. y /\ z e. w) -> (w e. y /\ z C_ w)))
9	8	ancomsd 485	. . . . . . . . . . . . 13 |- ((A e. On /\ y C_ A) -> ((z e. w /\ w e. y) -> (w e. y /\ z C_ w)))
10	9	eximdv 1669	. . . . . . . . . . . 12 |- ((A e. On /\ y C_ A) -> (E.w(z e. w /\ w e. y) -> E.w(w e. y /\ z C_ w)))
11		eluni 3180	. . . . . . . . . . . 12 |- (z e. U.y <-> E.w(z e. w /\ w e. y))
12		df-rex 2110	. . . . . . . . . . . 12 |- (E.w e. y z C_ w <-> E.w(w e. y /\ z C_ w))
13	10, 11, 12	3imtr4g 612	. . . . . . . . . . 11 |- ((A e. On /\ y C_ A) -> (z e. U.y -> E.w e. y z C_ w))
14	13	ralimdv 2172	. . . . . . . . . 10 |- ((A e. On /\ y C_ A) -> (A.z e. A z e. U.y -> A.z e. A E.w e. y z C_ w))
15		dfss3 2611	. . . . . . . . . 10 |- (A C_ U.y <-> A.z e. A z e. U.y)
16	14, 15	syl5ib 223	. . . . . . . . 9 |- ((A e. On /\ y C_ A) -> (A C_ U.y -> A.z e. A E.w e. y z C_ w))
17	16	ex 402	. . . . . . . 8 |- (A e. On -> (y C_ A -> (A C_ U.y -> A.z e. A E.w e. y z C_ w)))
18	17	imdistand 493	. . . . . . 7 |- (A e. On -> ((y C_ A /\ A C_ U.y) -> (y C_ A /\ A.z e. A E.w e. y z C_ w)))
19	18	anim2d 620	. . . . . 6 |- (A e. On -> ((x = (card` y) /\ (y C_ A /\ A C_ U.y)) -> (x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
20	19	eximdv 1669	. . . . 5 |- (A e. On -> (E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y)) -> E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
21	20	ss2abdv 2680	. . . 4 |- (A e. On -> {x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))} C_ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
22		intss 3239	. . . 4 |- ({x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))} C_ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))})
23	21, 22	syl 12	. . 3 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))})
24	1, 23	eqsstrd 2651	. 2 |- (A e. On -> (cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))})
25		0ss 2900	. . 3 |- (/) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))}
26		cffnon 6055	. . . . . . . 8 |- cf Fn On
27		fndm 4512	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
28	26, 27	ax-mp 7	. . . . . . 7 |- dom cf = On
29	28	eleq2i 1961	. . . . . 6 |- (A e. dom cf <-> A e. On)
30	29	notbii 204	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
31		ndmfv 4702	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
32	30, 31	sylbir 218	. . . 4 |- (-. A e. On -> (cf` A) = (/))
33	32	sseq1d 2644	. . 3 |- (-. A e. On -> ((cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))} <-> (/) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))}))
34	25, 33	mpbiri 211	. 2 |- (-. A e. On -> (cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))})
35	24, 34	pm2.61i 140	1 |- (cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))}

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214  Oncon0 3657  dom cdm 3986   Fn wfn 3993  ` cfv 3998  cardccrd 5859  cfccf 5861

This theorem is referenced by:  cflim 6057  cf0 6058

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  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-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-cf 5864

Copyright terms: Public domain