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Theorem uzdisj 10185
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 3347 . . . . . . 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 9630 . . . . . 6 |- ( k e. ( ZZ>= ` N ) -> N <_ k )
42, 3syl 14 . . . . 5 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> N <_ k )
5 eluzel2 9623 . . . . . . 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 9627 . . . . . . 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 9399 . . . . . 6 |- ( ( N e. ZZ /\ k e. ZZ ) -> ( N <_ k <-> ( N - 1 ) < k ) )
106, 8, 9syl2anc 411 . . . . 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 10120 . . . . . 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 9465 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. RR )
16 peano2zm 9381 . . . . . . . 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 9465 . . . . . 6 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> ( N - 1 ) e. RR )
1915, 18lenltd 8161 . . . . 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 621 . . 3 |- ( k e. ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) -> k e. (/) )
2221ssriv 3188 . 2 |- ( ( M ... ( N - 1 ) ) i^i ( ZZ>= ` N ) ) C_ (/)
23 ss0 3492 . 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
Syntax hints:   -. wn 3   <-> wb 105   = wceq 1364   e. wcel 2167   i^i cin 3156   C_ wss 3157   (/)c0 3451   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1c1 7897   < clt 8078   <_ cle 8079   - cmin 8214   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101
This theorem is referenced by:  2prm  12320
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