Theorem nn0disj 10334

Description: The first N + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)

Assertion

Ref	Expression
nn0disj	|- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) = (/)

Proof of Theorem nn0disj

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3387	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) <-> ( k e. ( 0 ... N ) /\ k e. ( ZZ>= ` ( N + 1 ) ) ) )
2	1	simprbi 275	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
3		eluzle 9734	. . . . . 6 |- ( k e. ( ZZ>= ` ( N + 1 ) ) -> ( N + 1 ) <_ k )
4	2, 3	syl 14	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( N + 1 ) <_ k )
5		eluzel2 9727	. . . . . . 7 |- ( k e. ( ZZ>= ` ( N + 1 ) ) -> ( N + 1 ) e. ZZ )
6	2, 5	syl 14	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( N + 1 ) e. ZZ )
7		eluzelz 9731	. . . . . . 7 |- ( k e. ( ZZ>= ` ( N + 1 ) ) -> k e. ZZ )
8	2, 7	syl 14	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. ZZ )
9		zlem1lt 9503	. . . . . 6 |- ( ( ( N + 1 ) e. ZZ /\ k e. ZZ ) -> ( ( N + 1 ) <_ k <-> ( ( N + 1 ) - 1 ) < k ) )
10	6, 8, 9	syl2anc 411	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( ( N + 1 ) <_ k <-> ( ( N + 1 ) - 1 ) < k ) )
11	4, 10	mpbid 147	. . . 4 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( ( N + 1 ) - 1 ) < k )
12	1	simplbi 274	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. ( 0 ... N ) )
13		elfzle2 10224	. . . . . 6 |- ( k e. ( 0 ... N ) -> k <_ N )
14	12, 13	syl 14	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k <_ N )
15	8	zred 9569	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. RR )
16		elfzel2 10219	. . . . . . . . . 10 |- ( k e. ( 0 ... N ) -> N e. ZZ )
17	16	adantr 276	. . . . . . . . 9 |- ( ( k e. ( 0 ... N ) /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ZZ )
18	1, 17	sylbi 121	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. ZZ )
19	18	zred 9569	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. RR )
20	15, 19	lenltd 8264	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. N < k ) )
21	18	zcnd 9570	. . . . . . . . . 10 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. CC )
22		pncan1 8523	. . . . . . . . . 10 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
23	21, 22	syl 14	. . . . . . . . 9 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( ( N + 1 ) - 1 ) = N )
24	23	eqcomd 2235	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N = ( ( N + 1 ) - 1 ) )
25	24	breq1d 4093	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( N < k <-> ( ( N + 1 ) - 1 ) < k ) )
26	25	notbid 671	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( -. N < k <-> -. ( ( N + 1 ) - 1 ) < k ) )
27	20, 26	bitrd 188	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. ( ( N + 1 ) - 1 ) < k ) )
28	14, 27	mpbid 147	. . . 4 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> -. ( ( N + 1 ) - 1 ) < k )
29	11, 28	pm2.21dd 623	. . 3 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. (/) )
30	29	ssriv 3228	. 2 |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) C_ (/)
31		ss0 3532	. 2 |- ( ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) C_ (/) -> ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) = (/) )
32	30, 31	ax-mp 5	1 |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) = (/)

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   i^i cin 3196   C_ wss 3197   (/)c0 3491   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   < clt 8181   <_ cle 8182   - cmin 8317   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205

This theorem is referenced by: (None)