Theorem fzdisj 8916

Description: Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)

Assertion

Ref	Expression
fzdisj	|- ( K < M -> ( ( J ... K ) i^i ( M ... N ) ) = (/) )

Proof of Theorem fzdisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3126	. . . 4 |- ( x e. ( ( J ... K ) i^i ( M ... N ) ) <-> ( x e. ( J ... K ) /\ x e. ( M ... N ) ) )
2		elfzel1 8889	. . . . . . . 8 |- ( x e. ( M ... N ) -> M e. ZZ )
3	2	adantl 262	. . . . . . 7 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> M e. ZZ )
4	3	zred 8360	. . . . . 6 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> M e. RR )
5		elfzelz 8890	. . . . . . . 8 |- ( x e. ( M ... N ) -> x e. ZZ )
6	5	zred 8360	. . . . . . 7 |- ( x e. ( M ... N ) -> x e. RR )
7	6	adantl 262	. . . . . 6 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> x e. RR )
8		elfzel2 8888	. . . . . . . 8 |- ( x e. ( J ... K ) -> K e. ZZ )
9	8	adantr 261	. . . . . . 7 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> K e. ZZ )
10	9	zred 8360	. . . . . 6 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> K e. RR )
11		elfzle1 8891	. . . . . . 7 |- ( x e. ( M ... N ) -> M <_ x )
12	11	adantl 262	. . . . . 6 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> M <_ x )
13		elfzle2 8892	. . . . . . 7 |- ( x e. ( J ... K ) -> x <_ K )
14	13	adantr 261	. . . . . 6 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> x <_ K )
15	4, 7, 10, 12, 14	letrd 7138	. . . . 5 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> M <_ K )
16	4, 10	lenltd 7134	. . . . 5 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> ( M <_ K <-> -. K < M ) )
17	15, 16	mpbid 135	. . . 4 |- ( ( x e. ( J ... K ) /\ x e. ( M ... N ) ) -> -. K < M )
18	1, 17	sylbi 114	. . 3 |- ( x e. ( ( J ... K ) i^i ( M ... N ) ) -> -. K < M )
19	18	con2i 557	. 2 |- ( K < M -> -. x e. ( ( J ... K ) i^i ( M ... N ) ) )
20	19	eq0rdv 3261	1 |- ( K < M -> ( ( J ... K ) i^i ( M ... N ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   i^i cin 2916   (/)c0 3224   class class class wbr 3764  (class class class)co 5512   RRcr 6888   < clt 7060   <_ cle 7061   ZZcz 8245   ...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-ltwlin 6997

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-nul 3225  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: (None)