Theorem cf0 6058

Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.

Assertion

Ref	Expression
cf0	|- (cf` (/)) = (/)

Proof of Theorem cf0

Step	Hyp	Ref	Expression
1		cfub 6056	. . 3 |- (cf` (/)) C_ |^|{x | E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y))}
2		0ss 2900	. . . . . . . . . . . . 13 |- (/) C_ U.y
3	2	biantru 793	. . . . . . . . . . . 12 |- (y C_ (/) <-> (y C_ (/) /\ (/) C_ U.y))
4		ss0b 2901	. . . . . . . . . . . 12 |- (y C_ (/) <-> y = (/))
5	3, 4	bitr3i 192	. . . . . . . . . . 11 |- ((y C_ (/) /\ (/) C_ U.y) <-> y = (/))
6	5	anbi2i 538	. . . . . . . . . 10 |- ((x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y)) <-> (x = (card` y) /\ y = (/)))
7		ancom 482	. . . . . . . . . 10 |- ((x = (card` y) /\ y = (/)) <-> (y = (/) /\ x = (card` y)))
8	6, 7	bitri 190	. . . . . . . . 9 |- ((x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y)) <-> (y = (/) /\ x = (card` y)))
9	8	exbii 1398	. . . . . . . 8 |- (E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y)) <-> E.y(y = (/) /\ x = (card` y)))
10		0ex 3446	. . . . . . . . 9 |- (/) e. _V
11		fveq2 4681	. . . . . . . . . 10 |- (y = (/) -> (card` y) = (card` (/)))
12	11	eqeq2d 1895	. . . . . . . . 9 |- (y = (/) -> (x = (card` y) <-> x = (card` (/))))
13	10, 12	ceqsexv 2325	. . . . . . . 8 |- (E.y(y = (/) /\ x = (card` y)) <-> x = (card` (/)))
14		card0 5869	. . . . . . . . 9 |- (card` (/)) = (/)
15	14	eqeq2i 1894	. . . . . . . 8 |- (x = (card` (/)) <-> x = (/))
16	9, 13, 15	3bitri 194	. . . . . . 7 |- (E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y)) <-> x = (/))
17	16	abbii 2006	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y))} = {x | x = (/)}
18		df-sn 3049	. . . . . 6 |- {(/)} = {x | x = (/)}
19	17, 18	eqtr4i 1911	. . . . 5 |- {x | E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y))} = {(/)}
20	19	inteqi 3218	. . . 4 |- |^|{x | E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y))} = |^|{(/)}
21	10	intsn 3252	. . . 4 |- |^|{(/)} = (/)
22	20, 21	eqtri 1908	. . 3 |- |^|{x | E.y(x = (card` y) /\ (y C_ (/) /\ (/) C_ U.y))} = (/)
23	1, 22	sseqtri 2649	. 2 |- (cf` (/)) C_ (/)
24		ss0b 2901	. 2 |- ((cf` (/)) C_ (/) <-> (cf` (/)) = (/))
25	23, 24	mpbi 206	1 |- (cf` (/)) = (/)

Syntax hints:   /\ wa 240   = wceq 1298  E.wex 1326  {cab 1871   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  |^|cint 3214  ` cfv 3998  cardccrd 5859  cfccf 5861

This theorem is referenced by:  cfeq0 6062

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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