Theorem caun0 23079

Description: A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

Assertion

Ref	Expression
caun0	|- ( ( D e. ( *Met ` X ) /\ F e. ( Cau ` D ) ) -> X =/= (/) )

Proof of Theorem caun0

Dummy variables j k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 11836	. . . 4 |- 1 e. RR+
2	1	ne0ii 3923	. . 3 |- RR+ =/= (/)
3		iscau2 23075	. . . 4 |- ( D e. ( *Met ` X ) -> ( F e. ( Cau ` D ) <-> ( F e. ( X ^pm CC ) /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) ) ) )
4	3	simplbda 654	. . 3 |- ( ( D e. ( *Met ` X ) /\ F e. ( Cau ` D ) ) -> A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) )
5		r19.2z 4060	. . 3 |- ( ( RR+ =/= (/) /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) ) -> E. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) )
6	2, 4, 5	sylancr 695	. 2 |- ( ( D e. ( *Met ` X ) /\ F e. ( Cau ` D ) ) -> E. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) )
7		uzid 11702	. . . . . 6 |- ( j e. ZZ -> j e. ( ZZ>= ` j ) )
8		ne0i 3921	. . . . . 6 |- ( j e. ( ZZ>= ` j ) -> ( ZZ>= ` j ) =/= (/) )
9		r19.2z 4060	. . . . . . 7 |- ( ( ( ZZ>= ` j ) =/= (/) /\ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) ) -> E. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) )
10	9	ex 450	. . . . . 6 |- ( ( ZZ>= ` j ) =/= (/) -> ( A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> E. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) ) )
11	7, 8, 10	3syl 18	. . . . 5 |- ( j e. ZZ -> ( A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> E. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) ) )
12		ne0i 3921	. . . . . . 7 |- ( ( F ` k ) e. X -> X =/= (/) )
13	12	3ad2ant2 1083	. . . . . 6 |- ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> X =/= (/) )
14	13	rexlimivw 3029	. . . . 5 |- ( E. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> X =/= (/) )
15	11, 14	syl6 35	. . . 4 |- ( j e. ZZ -> ( A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> X =/= (/) ) )
16	15	rexlimiv 3027	. . 3 |- ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> X =/= (/) )
17	16	rexlimivw 3029	. 2 |- ( E. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D ( F ` j ) ) < x ) -> X =/= (/) )
18	6, 17	syl 17	1 |- ( ( D e. ( *Met ` X ) /\ F e. ( Cau ` D ) ) -> X =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   1c1 9937   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   Caucca 23051

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-psmet 19738  df-xmet 19739  df-bl 19741  df-cau 23054

This theorem is referenced by:  cmetcau  23087