Theorem uzfissfz 39542

Description: For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
uzfissfz.m	|- ( ph -> M e. ZZ )
uzfissfz.z	|- Z = ( ZZ>= ` M )
uzfissfz.a	|- ( ph -> A C_ Z )
uzfissfz.fi	|- ( ph -> A e. Fin )

Assertion

Ref	Expression
uzfissfz	|- ( ph -> E. k e. Z A C_ ( M ... k ) )

Distinct variable groups:   A, k   k, M   k, Z

Allowed substitution hint:   ph( k)

Proof of Theorem uzfissfz

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

Step	Hyp	Ref	Expression
1		uzfissfz.m	. . . . . 6 |- ( ph -> M e. ZZ )
2		uzid 11702	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 17	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
4		uzfissfz.z	. . . . . . 7 |- Z = ( ZZ>= ` M )
5	4	a1i 11	. . . . . 6 |- ( ph -> Z = ( ZZ>= ` M ) )
6	5	eqcomd 2628	. . . . 5 |- ( ph -> ( ZZ>= ` M ) = Z )
7	3, 6	eleqtrd 2703	. . . 4 |- ( ph -> M e. Z )
8	7	adantr 481	. . 3 |- ( ( ph /\ A = (/) ) -> M e. Z )
9		id 22	. . . . 5 |- ( A = (/) -> A = (/) )
10		0ss 3972	. . . . . 6 |- (/) C_ ( M ... M )
11	10	a1i 11	. . . . 5 |- ( A = (/) -> (/) C_ ( M ... M ) )
12	9, 11	eqsstrd 3639	. . . 4 |- ( A = (/) -> A C_ ( M ... M ) )
13	12	adantl 482	. . 3 |- ( ( ph /\ A = (/) ) -> A C_ ( M ... M ) )
14		oveq2 6658	. . . . 5 |- ( k = M -> ( M ... k ) = ( M ... M ) )
15	14	sseq2d 3633	. . . 4 |- ( k = M -> ( A C_ ( M ... k ) <-> A C_ ( M ... M ) ) )
16	15	rspcev 3309	. . 3 |- ( ( M e. Z /\ A C_ ( M ... M ) ) -> E. k e. Z A C_ ( M ... k ) )
17	8, 13, 16	syl2anc 693	. 2 |- ( ( ph /\ A = (/) ) -> E. k e. Z A C_ ( M ... k ) )
18		uzfissfz.a	. . . . 5 |- ( ph -> A C_ Z )
19	18	adantr 481	. . . 4 |- ( ( ph /\ -. A = (/) ) -> A C_ Z )
20		uzssz 11707	. . . . . . . . 9 |- ( ZZ>= ` M ) C_ ZZ
21	4, 20	eqsstri 3635	. . . . . . . 8 |- Z C_ ZZ
22	21	a1i 11	. . . . . . 7 |- ( ph -> Z C_ ZZ )
23	18, 22	sstrd 3613	. . . . . 6 |- ( ph -> A C_ ZZ )
24	23	adantr 481	. . . . 5 |- ( ( ph /\ -. A = (/) ) -> A C_ ZZ )
25	9	necon3bi 2820	. . . . . 6 |- ( -. A = (/) -> A =/= (/) )
26	25	adantl 482	. . . . 5 |- ( ( ph /\ -. A = (/) ) -> A =/= (/) )
27		uzfissfz.fi	. . . . . 6 |- ( ph -> A e. Fin )
28	27	adantr 481	. . . . 5 |- ( ( ph /\ -. A = (/) ) -> A e. Fin )
29		suprfinzcl 11492	. . . . 5 |- ( ( A C_ ZZ /\ A =/= (/) /\ A e. Fin ) -> sup ( A , RR , < ) e. A )
30	24, 26, 28, 29	syl3anc 1326	. . . 4 |- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. A )
31	19, 30	sseldd 3604	. . 3 |- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. Z )
32	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> M e. ZZ )
33	21, 31	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ -. A = (/) ) -> sup ( A , RR , < ) e. ZZ )
34	33	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> sup ( A , RR , < ) e. ZZ )
35	24	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. ZZ )
36	32, 34, 35	3jca 1242	. . . . . . 7 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ j e. ZZ ) )
37	18	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ j e. A ) -> j e. Z )
38	4	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ j e. A ) -> Z = ( ZZ>= ` M ) )
39	37, 38	eleqtrd 2703	. . . . . . . . 9 |- ( ( ph /\ j e. A ) -> j e. ( ZZ>= ` M ) )
40		eluzle 11700	. . . . . . . . 9 |- ( j e. ( ZZ>= ` M ) -> M <_ j )
41	39, 40	syl 17	. . . . . . . 8 |- ( ( ph /\ j e. A ) -> M <_ j )
42	41	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> M <_ j )
43		zssre 11384	. . . . . . . . . 10 |- ZZ C_ RR
44	23, 43	syl6ss 3615	. . . . . . . . 9 |- ( ph -> A C_ RR )
45	44	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> A C_ RR )
46	26	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> A =/= (/) )
47		fimaxre2 10969	. . . . . . . . . 10 |- ( ( A C_ RR /\ A e. Fin ) -> E. x e. RR A. y e. A y <_ x )
48	44, 27, 47	syl2anc 693	. . . . . . . . 9 |- ( ph -> E. x e. RR A. y e. A y <_ x )
49	48	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> E. x e. RR A. y e. A y <_ x )
50		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. A )
51		suprub 10984	. . . . . . . 8 |- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ j e. A ) -> j <_ sup ( A , RR , < ) )
52	45, 46, 49, 50, 51	syl31anc 1329	. . . . . . 7 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j <_ sup ( A , RR , < ) )
53	36, 42, 52	jca32 558	. . . . . 6 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ j e. ZZ ) /\ ( M <_ j /\ j <_ sup ( A , RR , < ) ) ) )
54		elfz2 12333	. . . . . 6 |- ( j e. ( M ... sup ( A , RR , < ) ) <-> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ j e. ZZ ) /\ ( M <_ j /\ j <_ sup ( A , RR , < ) ) ) )
55	53, 54	sylibr 224	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ j e. A ) -> j e. ( M ... sup ( A , RR , < ) ) )
56	55	ralrimiva 2966	. . . 4 |- ( ( ph /\ -. A = (/) ) -> A. j e. A j e. ( M ... sup ( A , RR , < ) ) )
57		dfss3 3592	. . . 4 |- ( A C_ ( M ... sup ( A , RR , < ) ) <-> A. j e. A j e. ( M ... sup ( A , RR , < ) ) )
58	56, 57	sylibr 224	. . 3 |- ( ( ph /\ -. A = (/) ) -> A C_ ( M ... sup ( A , RR , < ) ) )
59		oveq2 6658	. . . . 5 |- ( k = sup ( A , RR , < ) -> ( M ... k ) = ( M ... sup ( A , RR , < ) ) )
60	59	sseq2d 3633	. . . 4 |- ( k = sup ( A , RR , < ) -> ( A C_ ( M ... k ) <-> A C_ ( M ... sup ( A , RR , < ) ) ) )
61	60	rspcev 3309	. . 3 |- ( ( sup ( A , RR , < ) e. Z /\ A C_ ( M ... sup ( A , RR , < ) ) ) -> E. k e. Z A C_ ( M ... k ) )
62	31, 58, 61	syl2anc 693	. 2 |- ( ( ph /\ -. A = (/) ) -> E. k e. Z A C_ ( M ... k ) )
63	17, 62	pm2.61dan 832	1 |- ( ph -> E. k e. Z A C_ ( M ... k ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   RRcr 9935   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

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  ax-pre-sup 10014

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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  sge0uzfsumgt  40661  sge0seq  40663  sge0reuz  40664  carageniuncllem2  40736  caratheodorylem2  40741