Theorem nn0disj 10494

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 3406	. . . . . . 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 9884	. . . . . 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 9876	. . . . . . 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 9881	. . . . . . 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 9651	. . . . . 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 10382	. . . . . 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 9718	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> k e. RR )
16		elfzel2 10376	. . . . . . . . . 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 9718	. . . . . . 7 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. RR )
20	15, 19	lenltd 8407	. . . . . 6 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> ( k <_ N <-> -. N < k ) )
21	18	zcnd 9719	. . . . . . . . . 10 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N e. CC )
22		pncan1 8667	. . . . . . . . . 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 2240	. . . . . . . 8 |- ( k e. ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) -> N = ( ( N + 1 ) - 1 ) )
25	24	breq1d 4124	. . . . . . 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 3246	. 2 |- ( ( 0 ... N ) i^i ( ZZ>= ` ( N + 1 ) ) ) C_ (/)
31		ss0 3553	. 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 2205   i^i cin 3213   C_ wss 3214   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362

This theorem is referenced by: (None)