Theorem ustn0 22024

Description: The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)

Assertion

Ref	Expression
ustn0	|- -. (/) e. U. ran UnifOn

Proof of Theorem ustn0

Dummy variables v u w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . . 5 |- -. ( x X. x ) e. (/)
2		0ex 4790	. . . . . 6 |- (/) e. _V
3		eleq2 2690	. . . . . 6 |- ( u = (/) -> ( ( x X. x ) e. u <-> ( x X. x ) e. (/) ) )
4	2, 3	elab 3350	. . . . 5 |- ( (/) e. { u | ( x X. x ) e. u } <-> ( x X. x ) e. (/) )
5	1, 4	mtbir 313	. . . 4 |- -. (/) e. { u | ( x X. x ) e. u }
6		vex 3203	. . . . . . 7 |- x e. _V
7		selpw 4165	. . . . . . . . . 10 |- ( u e. ~P ~P ( x X. x ) <-> u C_ ~P ( x X. x ) )
8	7	abbii 2739	. . . . . . . . 9 |- { u | u e. ~P ~P ( x X. x ) } = { u | u C_ ~P ( x X. x ) }
9		abid2 2745	. . . . . . . . . 10 |- { u | u e. ~P ~P ( x X. x ) } = ~P ~P ( x X. x )
10	6, 6	xpex 6962	. . . . . . . . . . . 12 |- ( x X. x ) e. _V
11	10	pwex 4848	. . . . . . . . . . 11 |- ~P ( x X. x ) e. _V
12	11	pwex 4848	. . . . . . . . . 10 |- ~P ~P ( x X. x ) e. _V
13	9, 12	eqeltri 2697	. . . . . . . . 9 |- { u | u e. ~P ~P ( x X. x ) } e. _V
14	8, 13	eqeltrri 2698	. . . . . . . 8 |- { u | u C_ ~P ( x X. x ) } e. _V
15		simp1 1061	. . . . . . . . 9 |- ( ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) -> u C_ ~P ( x X. x ) )
16	15	ss2abi 3674	. . . . . . . 8 |- { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } C_ { u | u C_ ~P ( x X. x ) }
17	14, 16	ssexi 4803	. . . . . . 7 |- { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } e. _V
18		df-ust 22004	. . . . . . . 8 |- UnifOn = ( x e. _V |-> { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } )
19	18	fvmpt2 6291	. . . . . . 7 |- ( ( x e. _V /\ { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } e. _V ) -> (UnifOn ` x ) = { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } )
20	6, 17, 19	mp2an 708	. . . . . 6 |- (UnifOn ` x ) = { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) }
21		simp2 1062	. . . . . . 7 |- ( ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) -> ( x X. x ) e. u )
22	21	ss2abi 3674	. . . . . 6 |- { u | ( u C_ ~P ( x X. x ) /\ ( x X. x ) e. u /\ A. v e. u ( A. w e. ~P ( x X. x ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` x ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) } C_ { u | ( x X. x ) e. u }
23	20, 22	eqsstri 3635	. . . . 5 |- (UnifOn ` x ) C_ { u | ( x X. x ) e. u }
24	23	sseli 3599	. . . 4 |- ( (/) e. (UnifOn ` x ) -> (/) e. { u | ( x X. x ) e. u } )
25	5, 24	mto 188	. . 3 |- -. (/) e. (UnifOn ` x )
26	25	nex 1731	. 2 |- -. E. x (/) e. (UnifOn ` x )
27	18	funmpt2 5927	. . . 4 |- Fun UnifOn
28		elunirn 6509	. . . 4 |- ( Fun UnifOn -> ( (/) e. U. ran UnifOn <-> E. x e. dom UnifOn (/) e. (UnifOn ` x ) ) )
29	27, 28	ax-mp 5	. . 3 |- ( (/) e. U. ran UnifOn <-> E. x e. dom UnifOn (/) e. (UnifOn ` x ) )
30		ustfn 22005	. . . . 5 |- UnifOn Fn _V
31		fndm 5990	. . . . 5 |- (UnifOn Fn _V -> dom UnifOn = _V )
32	30, 31	ax-mp 5	. . . 4 |- dom UnifOn = _V
33	32	rexeqi 3143	. . 3 |- ( E. x e. dom UnifOn (/) e. (UnifOn ` x ) <-> E. x e. _V (/) e. (UnifOn ` x ) )
34		rexv 3220	. . 3 |- ( E. x e. _V (/) e. (UnifOn ` x ) <-> E. x (/) e. (UnifOn ` x ) )
35	29, 33, 34	3bitri 286	. 2 |- ( (/) e. U. ran UnifOn <-> E. x (/) e. (UnifOn ` x ) )
36	26, 35	mtbir 313	1 |- -. (/) e. U. ran UnifOn

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  UnifOncust 22003

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-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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ust 22004

This theorem is referenced by: (None)