Theorem cf0 4910

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 4908	. . 3 |- (cf` (/)) (_ |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))}
2		0ss 2301	. . . . . . . . . . . . 13 |- (/) (_ U.y
3	2	biantru 724	. . . . . . . . . . . 12 |- (y (_ (/) <-> (y (_ (/) /\ (/) (_ U.y))
4		ss0b 2302	. . . . . . . . . . . 12 |- (y (_ (/) <-> y = (/))
5	3, 4	bitr3 175	. . . . . . . . . . 11 |- ((y (_ (/) /\ (/) (_ U.y) <-> y = (/))
6	5	anbi2i 480	. . . . . . . . . 10 |- ((x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> (x = (card` y) /\ y = (/)))
7		ancom 435	. . . . . . . . . 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 1051	. . . . . . . 8 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> E.y(y = (/) /\ x = (card` y)))
10		0ex 2711	. . . . . . . . 9 |- (/) e. V
11		fveq2 3724	. . . . . . . . . 10 |- (y = (/) -> (card` y) = (card` (/)))
12	11	eqeq2d 1486	. . . . . . . . 9 |- (y = (/) -> (x = (card` y) <-> x = (card` (/))))
13	10, 12	ceqsexv 1835	. . . . . . . 8 |- (E.y(y = (/) /\ x = (card` y)) <-> x = (card` (/)))
14		card0 4823	. . . . . . . . 9 |- (card` (/)) = (/)
15	14	eqeq2i 1485	. . . . . . . 8 |- (x = (card` (/)) <-> x = (/))
16	9, 13, 15	3bitr 177	. . . . . . 7 |- (E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y)) <-> x = (/))
17	16	abbii 1575	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {x | x = (/)}
18		df-sn 2412	. . . . . 6 |- {(/)} = {x | x = (/)}
19	17, 18	eqtr4 1498	. . . . 5 |- {x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = {(/)}
20	19	inteqi 2537	. . . 4 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = |^|{(/)}
21	10	intsn 2564	. . . 4 |- |^|{(/)} = (/)
22	20, 21	eqtr 1495	. . 3 |- |^|{x | E.y(x = (card` y) /\ (y (_ (/) /\ (/) (_ U.y))} = (/)
23	1, 22	sseqtr 2093	. 2 |- (cf` (/)) (_ (/)
24		ss0b 2302	. 2 |- ((cf` (/)) (_ (/) <-> (cf` (/)) = (/))
25	23, 24	mpbi 189	1 |- (cf` (/)) = (/)

Syntax hints:   /\ wa 223   = wceq 956  E.wex 980  {cab 1463   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  |^|cint 2533  ` cfv 3182  cardccrd 4813  cfccf 4815

This theorem is referenced by:  cfeq0 4914

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-card 4816  df-cf 4818

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