ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uzdisj Unicode version
Theorem uzdisj 10500
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.
StepHypRef Expression
1 elin 3412 . . . . . . 7 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) <-> ( k e. ( M ... ( N - 1 ) ) /\ k e. ( ZZ>= ` N ) ) )
21simprbi 275 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` N ) )
3 eluzle 9934 . . . . . 6 |- ( k e. ( ZZ>= ` N ) -> N <_ k )
42, 3syl 14 . . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> N <_ k )
5 eluzel2 9926 . . . . . . 7 |- ( k e. ( ZZ>= ` N ) -> N e. ZZ )
62, 5syl 14 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> N e. ZZ )
7 eluzelz 9931 . . . . . . 7 |- ( k e. ( ZZ>= ` N ) -> k e. ZZ )
82, 7syl 14 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ZZ )
9 zlem1lt 9701 . . . . . 6 |- ( ( N e. ZZ /\ k e. ZZ ) -> ( N <_ k <-> ( N - 1 ) < k ) )
106, 8, 9syl2anc 415 . . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N <_ k <-> ( N - 1 ) < k ) )
114, 10mpbid 147 . . . 4 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) < k )
121simplbi 274 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. ( M ... ( N - 1 ) ) )
13 elfzle2 10432 . . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k <_ ( N - 1 ) )
1412, 13syl 14 . . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k <_ ( N - 1 ) )
158zred 9768 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. RR )
16 peano2zm 9682 . . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
176, 16syl 14 . . . . . . 7 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. ZZ )
1817zred 9768 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. RR )
1915, 18lenltd 8444 . . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( k <_ ( N - 1 ) <-> -. ( N - 1 ) < k ) )
2014, 19mpbid 147 . . . 4 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> -. ( N - 1 ) < k )
2111, 20pm2.21dd 629 . . 3 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. (/) )
2221ssriv 3252 . 2 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/)
23 ss0 3563 . 2 |- ( ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/) -> ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/) )
2422, 23ax-mp 5 1 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) = (/)
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   <-> wb 105   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   (/)c0 3520   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1c1 8180   < clt 8360   <_ cle 8361   - cmin 8497   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412
This theorem is used by:  2prm  12905
  Copyright terms: Public domain W3C validator