Theorem fzopth 9079

Description: A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fzopth	|- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )

Proof of Theorem fzopth

Step	Hyp	Ref	Expression
1		eluzfz1 9050	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
2	1	adantr 270	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( M ... N ) )
3		simpr 108	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M ... N ) = ( J ... K ) )
4	2, 3	eleqtrd 2157	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( J ... K ) )
5		elfzuz 9041	. . . . . . 7 |- ( M e. ( J ... K ) -> M e. ( ZZ>= ` J ) )
6		uzss 8639	. . . . . . 7 |- ( M e. ( ZZ>= ` J ) -> ( ZZ>= ` M ) C_ ( ZZ>= ` J ) )
7	4, 5, 6	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) C_ ( ZZ>= ` J ) )
8		elfzuz2 9048	. . . . . . . . 9 |- ( M e. ( J ... K ) -> K e. ( ZZ>= ` J ) )
9		eluzfz1 9050	. . . . . . . . 9 |- ( K e. ( ZZ>= ` J ) -> J e. ( J ... K ) )
10	4, 8, 9	3syl 17	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( J ... K ) )
11	10, 3	eleqtrrd 2158	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( M ... N ) )
12		elfzuz 9041	. . . . . . 7 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
13		uzss 8639	. . . . . . 7 |- ( J e. ( ZZ>= ` M ) -> ( ZZ>= ` J ) C_ ( ZZ>= ` M ) )
14	11, 12, 13	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` J ) C_ ( ZZ>= ` M ) )
15	7, 14	eqssd 3016	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) = ( ZZ>= ` J ) )
16		eluzel2 8624	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 270	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ZZ )
18		uz11 8641	. . . . . 6 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` J ) <-> M = J ) )
19	17, 18	syl 14	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ( ZZ>= ` M ) = ( ZZ>= ` J ) <-> M = J ) )
20	15, 19	mpbid 145	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M = J )
21		eluzfz2 9051	. . . . . . . . 9 |- ( K e. ( ZZ>= ` J ) -> K e. ( J ... K ) )
22	4, 8, 21	3syl 17	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( J ... K ) )
23	22, 3	eleqtrrd 2158	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( M ... N ) )
24		elfzuz3 9042	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
25		uzss 8639	. . . . . . 7 |- ( N e. ( ZZ>= ` K ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` K ) )
26	23, 24, 25	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` K ) )
27		eluzfz2 9051	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
28	27	adantr 270	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( M ... N ) )
29	28, 3	eleqtrd 2157	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( J ... K ) )
30		elfzuz3 9042	. . . . . . 7 |- ( N e. ( J ... K ) -> K e. ( ZZ>= ` N ) )
31		uzss 8639	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` N ) )
32	29, 30, 31	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` N ) )
33	26, 32	eqssd 3016	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` K ) )
34		eluzelz 8628	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
35	34	adantr 270	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ZZ )
36		uz11 8641	. . . . . 6 |- ( N e. ZZ -> ( ( ZZ>= ` N ) = ( ZZ>= ` K ) <-> N = K ) )
37	35, 36	syl 14	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ( ZZ>= ` N ) = ( ZZ>= ` K ) <-> N = K ) )
38	33, 37	mpbid 145	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N = K )
39	20, 38	jca 300	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M = J /\ N = K ) )
40	39	ex 113	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) -> ( M = J /\ N = K ) ) )
41		oveq12 5541	. 2 |- ( ( M = J /\ N = K ) -> ( M ... N ) = ( J ... K ) )
42	40, 41	impbid1 140	1 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   C_ wss 2973   ` cfv 4922  (class class class)co 5532   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-apti 7091

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-neg 7282  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  2ffzeq  9151