Theorem ufilen 21734

Description: Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)

Assertion

Ref	Expression
ufilen	|- ( om ~<_ X -> E. f e. ( UFil ` X ) A. x e. f x ~~ X )

Distinct variable group:   x, f, X

Proof of Theorem ufilen

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . . 6 |- Rel ~<_
2	1	brrelex2i 5159	. . . . 5 |- ( om ~<_ X -> X e. _V )
3		numth3 9292	. . . . 5 |- ( X e. _V -> X e. dom card )
4	2, 3	syl 17	. . . 4 |- ( om ~<_ X -> X e. dom card )
5		csdfil 21698	. . . 4 |- ( ( X e. dom card /\ om ~<_ X ) -> { y e. ~P X | ( X \ y ) ~< X } e. ( Fil ` X ) )
6	4, 5	mpancom 703	. . 3 |- ( om ~<_ X -> { y e. ~P X | ( X \ y ) ~< X } e. ( Fil ` X ) )
7		filssufil 21716	. . 3 |- ( { y e. ~P X | ( X \ y ) ~< X } e. ( Fil ` X ) -> E. f e. ( UFil ` X ) { y e. ~P X | ( X \ y ) ~< X } C_ f )
8	6, 7	syl 17	. 2 |- ( om ~<_ X -> E. f e. ( UFil ` X ) { y e. ~P X | ( X \ y ) ~< X } C_ f )
9		elfvex 6221	. . . . . . 7 |- ( f e. ( UFil ` X ) -> X e. _V )
10	9	ad2antlr 763	. . . . . 6 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> X e. _V )
11		ufilfil 21708	. . . . . . . 8 |- ( f e. ( UFil ` X ) -> f e. ( Fil ` X ) )
12		filelss 21656	. . . . . . . 8 |- ( ( f e. ( Fil ` X ) /\ x e. f ) -> x C_ X )
13	11, 12	sylan 488	. . . . . . 7 |- ( ( f e. ( UFil ` X ) /\ x e. f ) -> x C_ X )
14	13	adantll 750	. . . . . 6 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> x C_ X )
15		ssdomg 8001	. . . . . 6 |- ( X e. _V -> ( x C_ X -> x ~<_ X ) )
16	10, 14, 15	sylc 65	. . . . 5 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> x ~<_ X )
17		filfbas 21652	. . . . . . . . 9 |- ( f e. ( Fil ` X ) -> f e. ( fBas ` X ) )
18	11, 17	syl 17	. . . . . . . 8 |- ( f e. ( UFil ` X ) -> f e. ( fBas ` X ) )
19	18	adantl 482	. . . . . . 7 |- ( ( om ~<_ X /\ f e. ( UFil ` X ) ) -> f e. ( fBas ` X ) )
20		fbncp 21643	. . . . . . 7 |- ( ( f e. ( fBas ` X ) /\ x e. f ) -> -. ( X \ x ) e. f )
21	19, 20	sylan 488	. . . . . 6 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> -. ( X \ x ) e. f )
22		difss 3737	. . . . . . . . . . . . . 14 |- ( X \ x ) C_ X
23		elpw2g 4827	. . . . . . . . . . . . . 14 |- ( X e. _V -> ( ( X \ x ) e. ~P X <-> ( X \ x ) C_ X ) )
24	22, 23	mpbiri 248	. . . . . . . . . . . . 13 |- ( X e. _V -> ( X \ x ) e. ~P X )
25	24	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> ( X \ x ) e. ~P X )
26		simp2 1062	. . . . . . . . . . . . . 14 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> x C_ X )
27		dfss4 3858	. . . . . . . . . . . . . 14 |- ( x C_ X <-> ( X \ ( X \ x ) ) = x )
28	26, 27	sylib 208	. . . . . . . . . . . . 13 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> ( X \ ( X \ x ) ) = x )
29		simp3 1063	. . . . . . . . . . . . 13 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> x ~< X )
30	28, 29	eqbrtrd 4675	. . . . . . . . . . . 12 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> ( X \ ( X \ x ) ) ~< X )
31		difeq2 3722	. . . . . . . . . . . . . 14 |- ( y = ( X \ x ) -> ( X \ y ) = ( X \ ( X \ x ) ) )
32	31	breq1d 4663	. . . . . . . . . . . . 13 |- ( y = ( X \ x ) -> ( ( X \ y ) ~< X <-> ( X \ ( X \ x ) ) ~< X ) )
33	32	elrab 3363	. . . . . . . . . . . 12 |- ( ( X \ x ) e. { y e. ~P X | ( X \ y ) ~< X } <-> ( ( X \ x ) e. ~P X /\ ( X \ ( X \ x ) ) ~< X ) )
34	25, 30, 33	sylanbrc 698	. . . . . . . . . . 11 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> ( X \ x ) e. { y e. ~P X | ( X \ y ) ~< X } )
35		ssel 3597	. . . . . . . . . . 11 |- ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> ( ( X \ x ) e. { y e. ~P X | ( X \ y ) ~< X } -> ( X \ x ) e. f ) )
36	34, 35	syl5com 31	. . . . . . . . . 10 |- ( ( X e. _V /\ x C_ X /\ x ~< X ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> ( X \ x ) e. f ) )
37	36	3expa 1265	. . . . . . . . 9 |- ( ( ( X e. _V /\ x C_ X ) /\ x ~< X ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> ( X \ x ) e. f ) )
38	37	impancom 456	. . . . . . . 8 |- ( ( ( X e. _V /\ x C_ X ) /\ { y e. ~P X | ( X \ y ) ~< X } C_ f ) -> ( x ~< X -> ( X \ x ) e. f ) )
39	38	con3d 148	. . . . . . 7 |- ( ( ( X e. _V /\ x C_ X ) /\ { y e. ~P X | ( X \ y ) ~< X } C_ f ) -> ( -. ( X \ x ) e. f -> -. x ~< X ) )
40	39	impancom 456	. . . . . 6 |- ( ( ( X e. _V /\ x C_ X ) /\ -. ( X \ x ) e. f ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> -. x ~< X ) )
41	10, 14, 21, 40	syl21anc 1325	. . . . 5 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> -. x ~< X ) )
42		bren2 7986	. . . . . 6 |- ( x ~~ X <-> ( x ~<_ X /\ -. x ~< X ) )
43	42	simplbi2 655	. . . . 5 |- ( x ~<_ X -> ( -. x ~< X -> x ~~ X ) )
44	16, 41, 43	sylsyld 61	. . . 4 |- ( ( ( om ~<_ X /\ f e. ( UFil ` X ) ) /\ x e. f ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> x ~~ X ) )
45	44	ralrimdva 2969	. . 3 |- ( ( om ~<_ X /\ f e. ( UFil ` X ) ) -> ( { y e. ~P X | ( X \ y ) ~< X } C_ f -> A. x e. f x ~~ X ) )
46	45	reximdva 3017	. 2 |- ( om ~<_ X -> ( E. f e. ( UFil ` X ) { y e. ~P X | ( X \ y ) ~< X } C_ f -> E. f e. ( UFil ` X ) A. x e. f x ~~ X ) )
47	8, 46	mpd 15	1 |- ( om ~<_ X -> E. f e. ( UFil ` X ) A. x e. f x ~~ X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   fBascfbas 19734   Filcfil 21649   UFilcufil 21703

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-oi 8415  df-card 8765  df-ac 8939  df-cda 8990  df-fbas 19743  df-fg 19744  df-fil 21650  df-ufil 21705

This theorem is referenced by: (None)