Theorem nn0disj 10435

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 3392	. . . . . . 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 9829	. . . . . 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 9821	. . . . . . 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 9826	. . . . . . 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 9597	. . . . . 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 10325	. . . . . 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 9663	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. RR )
16		elfzel2 10320	. . . . . . . . . 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 9663	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. RR )
20	15, 19	lenltd 8356	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. N < k ) )
21	18	zcnd 9664	. . . . . . . . . 10 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. CC )
22		pncan1 8615	. . . . . . . . . 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 2237	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N = ( ( N + 1 ) - 1 ) )
25	24	breq1d 4103	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( N < k <-> ( ( N + 1 ) - 1 ) < k ) )
26	25	notbid 673	. . . . . 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 625	. . 3 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. (/) )
30	29	ssriv 3232	. 2 |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) C_ (/)
31		ss0 3537	. 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 1398   e. wcel 2202   i^i cin 3200   C_ wss 3201   (/)c0 3496   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   1c1 8093   + caddc 8095   < clt 8273   <_ cle 8274   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by: (None)