Theorem fzopth 8924

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 8895	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
2	1	adantr 261	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( M ... N ) )
3		simpr 103	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M ... N ) = ( J ... K ) )
4	2, 3	eleqtrd 2116	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( J ... K ) )
5		elfzuz 8886	. . . . . . 7 |- ( M e. ( J ... K ) -> M e. ( ZZ>= ` J ) )
6		uzss 8493	. . . . . . 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 8893	. . . . . . . . 9 |- ( M e. ( J ... K ) -> K e. ( ZZ>= ` J ) )
9		eluzfz1 8895	. . . . . . . . 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 2117	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( M ... N ) )
12		elfzuz 8886	. . . . . . 7 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
13		uzss 8493	. . . . . . 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 2962	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) = ( ZZ>= ` J ) )
16		eluzel2 8478	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 261	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ZZ )
18		uz11 8495	. . . . . 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 135	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M = J )
21		eluzfz2 8896	. . . . . . . . 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 2117	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( M ... N ) )
24		elfzuz3 8887	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
25		uzss 8493	. . . . . . 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 8896	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
28	27	adantr 261	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( M ... N ) )
29	28, 3	eleqtrd 2116	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( J ... K ) )
30		elfzuz3 8887	. . . . . . 7 |- ( N e. ( J ... K ) -> K e. ( ZZ>= ` N ) )
31		uzss 8493	. . . . . . 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 2962	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` K ) )
34		eluzelz 8482	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
35	34	adantr 261	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ZZ )
36		uz11 8495	. . . . . 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 135	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N = K )
39	20, 38	jca 290	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M = J /\ N = K ) )
40	39	ex 108	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) -> ( M = J /\ N = K ) ) )
41		oveq12 5521	. 2 |- ( ( M = J /\ N = K ) -> ( M ... N ) = ( J ... K ) )
42	40, 41	impbid1 130	1 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   C_ wss 2917   ` cfv 4902  (class class class)co 5512   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-apti 6999

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-neg 7185  df-z 8246  df-uz 8474  df-fz 8875

This theorem is referenced by:  2ffzeq  8998