Theorem cf0 3705

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 3703	. . 3 |- (cf` (/)) (_ |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))}
2		0ss 1725	. . . . . . . . . . . . 13 |- (/) (_ U.y
3	2	biantru 543	. . . . . . . . . . . 12 |- (y (_ (/) <-> (y (_ (/) /\ (/) (_ U.y))
4		ss0b 1726	. . . . . . . . . . . 12 |- (y (_ (/) <-> y = (/))
5	3, 4	bitr3 153	. . . . . . . . . . 11 |- ((y (_ (/) /\ (/) (_ U.y) <-> y = (/))
6	5	anbi2i 367	. . . . . . . . . 10 |- ((x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> (x = (card` y) /\ y = (/)))
7		ancom 333	. . . . . . . . . 10 |- ((x = (card` y) /\ y = (/)) <-> (y = (/) /\ x = (card` y)))
8	6, 7	bitr 151	. . . . . . . . 9 |- ((x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> (y = (/) /\ x = (card` y)))
9	8	biex 733	. . . . . . . 8 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> E.y(y = (/) /\ x = (card` y)))
10		0ex 1745	. . . . . . . . 9 |- (/) e. V
11		fveq2 2832	. . . . . . . . . 10 |- (y = (/) -> (card` y) = (card` (/)))
12	11	cleq2d 1112	. . . . . . . . 9 |- (y = (/) -> (x = (card` y) <-> x = (card` (/))))
13	10, 12	ceqsexv 1371	. . . . . . . 8 |- (E.y(y = (/) /\ x = (card` y)) <-> x = (card` (/)))
14		card0 3630	. . . . . . . . 9 |- (card` (/)) = (/)
15	14	cleq2i 1111	. . . . . . . 8 |- (x = (card` (/)) <-> x = (/))
16	9, 13, 15	3bitr 155	. . . . . . 7 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> x = (/))
17	16	biabi 1181	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {x | x = (/)}
18		df-sn 1811	. . . . . 6 |- {(/)} = {x | x = (/)}
19	17, 18	eqtr4 1122	. . . . 5 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {(/)}
20	19	inteqi 1969	. . . 4 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = |^|{(/)}
21	10	intsn 1991	. . . 4 |- |^|{(/)} = (/)
22	20, 21	eqtr 1119	. . 3 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = (/)
23	1, 22	sseqtr 1532	. 2 |- (cf` (/)) (_ (/)
24		ss0b 1726	. 2 |- ((cf` (/)) (_ (/) <-> (cf` (/)) = (/))
25	23, 24	mpbi 164	1 |- (cf` (/)) = (/)

Syntax hints:   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   (_ wss 1487  (/)c0 1707  {csn 1808  U.cuni 1919  |^|cint 1965  ` cfv 2422  cardccrd 3620  cfccf 3622

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-card 3623  df-cf 3625

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