Theorem cf0 4922

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 4920	. . 3 |- (cf` (/)) (_ |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))}
2		0ss 2305	. . . . . . . . . . . . 13 |- (/) (_ U.y
3	2	biantru 726	. . . . . . . . . . . 12 |- (y (_ (/) <-> (y (_ (/) /\ (/) (_ U.y))
4		ss0b 2306	. . . . . . . . . . . 12 |- (y (_ (/) <-> y = (/))
5	3, 4	bitr3 175	. . . . . . . . . . 11 |- ((y (_ (/) /\ (/) (_ U.y) <-> y = (/))
6	5	anbi2i 482	. . . . . . . . . 10 |- ((x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> (x = (card` y) /\ y = (/)))
7		ancom 437	. . . . . . . . . 10 |- ((x = (card` y) /\ y = (/)) <-> (y = (/) /\ x = (card` y)))
8	6, 7	bitr 173	. . . . . . . . 9 |- ((x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> (y = (/) /\ x = (card` y)))
9	8	exbii 1053	. . . . . . . 8 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> E.y(y = (/) /\ x = (card` y)))
10		0ex 2716	. . . . . . . . 9 |- (/) e. V
11		fveq2 3730	. . . . . . . . . 10 |- (y = (/) -> (card` y) = (card` (/)))
12	11	eqeq2d 1489	. . . . . . . . 9 |- (y = (/) -> (x = (card` y) <-> x = (card` (/))))
13	10, 12	ceqsexv 1838	. . . . . . . 8 |- (E.y(y = (/) /\ x = (card` y)) <-> x = (card` (/)))
14		card0 4833	. . . . . . . . 9 |- (card` (/)) = (/)
15	14	eqeq2i 1488	. . . . . . . 8 |- (x = (card` (/)) <-> x = (/))
16	9, 13, 15	3bitr 177	. . . . . . 7 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> x = (/))
17	16	abbii 1578	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {x | x = (/)}
18		df-sn 2416	. . . . . 6 |- {(/)} = {x | x = (/)}
19	17, 18	eqtr4 1501	. . . . 5 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {(/)}
20	19	inteqi 2541	. . . 4 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = |^|{(/)}
21	10	intsn 2568	. . . 4 |- |^|{(/)} = (/)
22	20, 21	eqtr 1498	. . 3 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = (/)
23	1, 22	sseqtr 2096	. 2 |- (cf` (/)) (_ (/)
24		ss0b 2306	. 2 |- ((cf` (/)) (_ (/) <-> (cf` (/)) = (/))
25	23, 24	mpbi 189	1 |- (cf` (/)) = (/)

Syntax hints:   /\ wa 223   = wceq 958  E.wex 982  {cab 1466   (_ wss 2050  (/)c0 2283  {csn 2413  U.cuni 2507  |^|cint 2537  ` cfv 3188  cardccrd 4823  cfccf 4825

This theorem is referenced by:  cfeq0 4926

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-en 4374  df-card 4826  df-cf 4828

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