Theorem uzdisj 10328

Description: The first N elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Assertion

Ref	Expression
uzdisj	|- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/)

Proof of Theorem uzdisj

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3390	. . . . . . 7 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) <-> ( k e. ( M ... ( N - 1 ) ) /\ k e. ( ZZ>= ` N ) ) )
2	1	simprbi 275	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` N ) )
3		eluzle 9768	. . . . . 6 |- ( k e. ( ZZ>= ` N ) -> N <_ k )
4	2, 3	syl 14	. . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> N <_ k )
5		eluzel2 9760	. . . . . . 7 |- ( k e. ( ZZ>= ` N ) -> N e. ZZ )
6	2, 5	syl 14	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> N e. ZZ )
7		eluzelz 9765	. . . . . . 7 |- ( k e. ( ZZ>= ` N ) -> k e. ZZ )
8	2, 7	syl 14	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ZZ )
9		zlem1lt 9536	. . . . . 6 |- ( ( N e. ZZ /\ k e. ZZ ) -> ( N <_ k <-> ( N - 1 ) < k ) )
10	6, 8, 9	syl2anc 411	. . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N <_ k <-> ( N - 1 ) < k ) )
11	4, 10	mpbid 147	. . . 4 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) < k )
12	1	simplbi 274	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ( M ... ( N - 1 ) ) )
13		elfzle2 10263	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k <_ ( N - 1 ) )
14	12, 13	syl 14	. . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k <_ ( N - 1 ) )
15	8	zred 9602	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. RR )
16		peano2zm 9517	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
17	6, 16	syl 14	. . . . . . 7 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. ZZ )
18	17	zred 9602	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. RR )
19	15, 18	lenltd 8297	. . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( k <_ ( N - 1 ) <-> -. ( N - 1 ) < k ) )
20	14, 19	mpbid 147	. . . 4 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> -. ( N - 1 ) < k )
21	11, 20	pm2.21dd 625	. . 3 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. (/) )
22	21	ssriv 3231	. 2 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/)
23		ss0 3535	. 2 |- ( ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/) -> ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/) )
24	22, 23	ax-mp 5	1 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/)

Syntax hints:   -. wn 3   <-> wb 105   = wceq 1397   e. wcel 2202   i^i cin 3199   C_ wss 3200   (/)c0 3494   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   1c1 8033   < clt 8214   <_ cle 8215   - cmin 8350   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244

This theorem is referenced by:  2prm  12700