Theorem cfle 6061

Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cfle	|- (cf` A) C_ A

Proof of Theorem cfle

Step	Hyp	Ref	Expression
1		cardonle 5868	. . 3 |- (A e. On -> (card` A) C_ A)
2		cflecard 6060	. . . 4 |- (cf` A) C_ (card` A)
3		sstr2 2623	. . . 4 |- ((cf` A) C_ (card` A) -> ((card` A) C_ A -> (cf` A) C_ A))
4	2, 3	ax-mp 7	. . 3 |- ((card` A) C_ A -> (cf` A) C_ A)
5	1, 4	syl 12	. 2 |- (A e. On -> (cf` A) C_ A)
6		0ss 2900	. . 3 |- (/) C_ A
7		cffnon 6055	. . . . . . . 8 |- cf Fn On
8		fndm 4512	. . . . . . . 8 |- (cf Fn On -> dom cf = On)
9	7, 8	ax-mp 7	. . . . . . 7 |- dom cf = On
10	9	eleq2i 1961	. . . . . 6 |- (A e. dom cf <-> A e. On)
11	10	notbii 204	. . . . 5 |- (-. A e. dom cf <-> -. A e. On)
12		ndmfv 4702	. . . . 5 |- (-. A e. dom cf -> (cf` A) = (/))
13	11, 12	sylbir 218	. . . 4 |- (-. A e. On -> (cf` A) = (/))
14	13	sseq1d 2644	. . 3 |- (-. A e. On -> ((cf` A) C_ A <-> (/) C_ A))
15	6, 14	mpbiri 211	. 2 |- (-. A e. On -> (cf` A) C_ A)
16	5, 15	pm2.61i 140	1 |- (cf` A) C_ A

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  Oncon0 3657  dom cdm 3986   Fn wfn 3993  ` cfv 3998  cardccrd 5859  cfccf 5861

This theorem is referenced by:  cfom 6064

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-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-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-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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427  df-card 5862  df-cf 5864

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