Theorem uzdisj 10413

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 3401	. . . . . . 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 9852	. . . . . 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 9844	. . . . . . 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 9849	. . . . . . 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 9620	. . . . . 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 10348	. . . . . 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 9686	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. RR )
16		peano2zm 9601	. . . . . . . 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 9686	. . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. RR )
19	15, 18	lenltd 8379	. . . . 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 3241	. 2 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/)
23		ss0 3546	. 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 1398   e. wcel 2203   i^i cin 3209   C_ wss 3210   (/)c0 3505   class class class wbr 4102   ` cfv 5343  (class class class)co 6041   1c1 8116   < clt 8296   <_ cle 8297   - cmin 8432   ZZcz 9563   ZZ>=cuz 9839   ...cfz 10328

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-distr 8219  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-cnre 8226  ax-pre-ltirr 8227  ax-pre-ltwlin 8228  ax-pre-lttrn 8229  ax-pre-ltadd 8231

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-pnf 8298  df-mnf 8299  df-xr 8300  df-ltxr 8301  df-le 8302  df-sub 8434  df-neg 8435  df-inn 9226  df-n0 9485  df-z 9564  df-uz 9840  df-fz 10329

This theorem is referenced by:  2prm  12802