Theorem nn0disj 10073

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 3305	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) <-> ( k e. ( 0 ... N ) /\ k e. ( ZZ>= ` ( N + 1 ) ) ) )
2	1	simprbi 273	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
3		eluzle 9478	. . . . . 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 9471	. . . . . . 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 9475	. . . . . . 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 9247	. . . . . 6 |- ( ( ( N + 1 ) e. ZZ /\ k e. ZZ ) -> ( ( N + 1 ) <_ k <-> ( ( N + 1 ) - 1 ) < k ) )
10	6, 8, 9	syl2anc 409	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( ( N + 1 ) <_ k <-> ( ( N + 1 ) - 1 ) < k ) )
11	4, 10	mpbid 146	. . . 4 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( ( N + 1 ) - 1 ) < k )
12	1	simplbi 272	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. ( 0 ... N ) )
13		elfzle2 9963	. . . . . 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 9313	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. RR )
16		elfzel2 9958	. . . . . . . . . 10 |- ( k e. ( 0 ... N ) -> N e. ZZ )
17	16	adantr 274	. . . . . . . . 9 |- ( ( k e. ( 0 ... N ) /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ZZ )
18	1, 17	sylbi 120	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. ZZ )
19	18	zred 9313	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. RR )
20	15, 19	lenltd 8016	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. N < k ) )
21	18	zcnd 9314	. . . . . . . . . 10 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. CC )
22		pncan1 8275	. . . . . . . . . 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 2171	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N = ( ( N + 1 ) - 1 ) )
25	24	breq1d 3992	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( N < k <-> ( ( N + 1 ) - 1 ) < k ) )
26	25	notbid 657	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( -. N < k <-> -. ( ( N + 1 ) - 1 ) < k ) )
27	20, 26	bitrd 187	. . . . 5 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. ( ( N + 1 ) - 1 ) < k ) )
28	14, 27	mpbid 146	. . . 4 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> -. ( ( N + 1 ) - 1 ) < k )
29	11, 28	pm2.21dd 610	. . 3 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. (/) )
30	29	ssriv 3146	. 2 |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) C_ (/)
31		ss0 3449	. 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 103   <-> wb 104   = wceq 1343   e. wcel 2136   i^i cin 3115   C_ wss 3116   (/)c0 3409   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   < clt 7933   <_ cle 7934   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by: (None)