Theorem cf0 9073

Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)

Assertion

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

Proof of Theorem cf0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfub 9071	. . 3 |- ( cf ` (/) ) C_ |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) }
2		0ss 3972	. . . . . . . . . . . . 13 |- (/) C_ U. y
3	2	biantru 526	. . . . . . . . . . . 12 |- ( y C_ (/) <-> ( y C_ (/) /\ (/) C_ U. y ) )
4		ss0b 3973	. . . . . . . . . . . 12 |- ( y C_ (/) <-> y = (/) )
5	3, 4	bitr3i 266	. . . . . . . . . . 11 |- ( ( y C_ (/) /\ (/) C_ U. y ) <-> y = (/) )
6	5	anbi2i 730	. . . . . . . . . 10 |- ( ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) <-> ( x = ( card ` y ) /\ y = (/) ) )
7		ancom 466	. . . . . . . . . 10 |- ( ( x = ( card ` y ) /\ y = (/) ) <-> ( y = (/) /\ x = ( card ` y ) ) )
8	6, 7	bitri 264	. . . . . . . . 9 |- ( ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) <-> ( y = (/) /\ x = ( card ` y ) ) )
9	8	exbii 1774	. . . . . . . 8 |- ( E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) <-> E. y ( y = (/) /\ x = ( card ` y ) ) )
10		0ex 4790	. . . . . . . . . 10 |- (/) e. _V
11		fveq2 6191	. . . . . . . . . . 11 |- ( y = (/) -> ( card ` y ) = ( card ` (/) ) )
12	11	eqeq2d 2632	. . . . . . . . . 10 |- ( y = (/) -> ( x = ( card ` y ) <-> x = ( card ` (/) ) ) )
13	10, 12	ceqsexv 3242	. . . . . . . . 9 |- ( E. y ( y = (/) /\ x = ( card ` y ) ) <-> x = ( card ` (/) ) )
14		card0 8784	. . . . . . . . . 10 |- ( card ` (/) ) = (/)
15	14	eqeq2i 2634	. . . . . . . . 9 |- ( x = ( card ` (/) ) <-> x = (/) )
16	13, 15	bitri 264	. . . . . . . 8 |- ( E. y ( y = (/) /\ x = ( card ` y ) ) <-> x = (/) )
17	9, 16	bitri 264	. . . . . . 7 |- ( E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) <-> x = (/) )
18	17	abbii 2739	. . . . . 6 |- { x | E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) } = { x | x = (/) }
19		df-sn 4178	. . . . . 6 |- { (/) } = { x | x = (/) }
20	18, 19	eqtr4i 2647	. . . . 5 |- { x | E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) } = { (/) }
21	20	inteqi 4479	. . . 4 |- |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) } = |^| { (/) }
22	10	intsn 4513	. . . 4 |- |^| { (/) } = (/)
23	21, 22	eqtri 2644	. . 3 |- |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ (/) /\ (/) C_ U. y ) ) } = (/)
24	1, 23	sseqtri 3637	. 2 |- ( cf ` (/) ) C_ (/)
25		ss0b 3973	. 2 |- ( ( cf ` (/) ) C_ (/) <-> ( cf ` (/) ) = (/) )
26	24, 25	mpbi 220	1 |- ( cf ` (/) ) = (/)

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   {cab 2608   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   |^|cint 4475   ` cfv 5888   cardccrd 8761   cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-card 8765  df-cf 8767

This theorem is referenced by:  cfeq0  9078  cflim2  9085  cfidm  9097  alephsing  9098  alephreg  9404  pwcfsdom  9405  rankcf  9599