ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uz11 Unicode version
Theorem uz11 9488
Description: The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
Assertion
Ref Expression
uz11 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )
Proof of Theorem uz11
StepHypRef Expression
1 uzid 9480 . . . . 5 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
2 eleq2 2230 . . . . . 6 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) <-> M e. ( ZZ>= ` N ) ) )
3 eluzel2 9471 . . . . . 6 |- ( M e. ( ZZ>= ` N ) -> N e. ZZ )
42, 3syl6bi 162 . . . . 5 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ( ZZ>= ` M ) -> N e. ZZ ) )
51, 4mpan9 279 . . . 4 |- ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> N e. ZZ )
6 uzid 9480 . . . . . . . . . . 11 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
7 eleq2 2230 . . . . . . . . . . 11 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ( ZZ>= ` M ) <-> N e. ( ZZ>= ` N ) ) )
86, 7syl5ibr 155 . . . . . . . . . 10 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> N e. ( ZZ>= ` M ) ) )
9 eluzle 9478 . . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
108, 9syl6 33 . . . . . . . . 9 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( N e. ZZ -> M <_ N ) )
111, 2syl5ib 153 . . . . . . . . . 10 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> M e. ( ZZ>= ` N ) ) )
12 eluzle 9478 . . . . . . . . . 10 |- ( M e. ( ZZ>= ` N ) -> N <_ M )
1311, 12syl6 33 . . . . . . . . 9 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( M e. ZZ -> N <_ M ) )
1410, 13anim12d 333 . . . . . . . 8 |- ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> ( ( N e. ZZ /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) ) )
1514impl 378 . . . . . . 7 |- ( ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) /\ M e. ZZ ) -> ( M <_ N /\ N <_ M ) )
1615ancoms 266 . . . . . 6 |- ( ( M e. ZZ /\ ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ N e. ZZ ) ) -> ( M <_ N /\ N <_ M ) )
1716anassrs 398 . . . . 5 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M <_ N /\ N <_ M ) )
18 zre 9195 . . . . . . 7 |- ( M e. ZZ -> M e. RR )
19 zre 9195 . . . . . . 7 |- ( N e. ZZ -> N e. RR )
20 letri3 7979 . . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
2118, 19, 20syl2an 287 . . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
2221adantlr 469 . . . . 5 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
2317, 22mpbird 166 . . . 4 |- ( ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) /\ N e. ZZ ) -> M = N )
245, 23mpdan 418 . . 3 |- ( ( M e. ZZ /\ ( ZZ>= ` M ) = ( ZZ>= ` N ) ) -> M = N )
2524ex 114 . 2 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) -> M = N ) )
26 fveq2 5486 . 2 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
2725, 26impbid1 141 1 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188   RRcr 7752   <_ cle 7934   ZZcz 9191   ZZ>=cuz 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865  ax-pre-apti 7868
This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-neg 8072  df-z 9192  df-uz 9467
This theorem is referenced by:  fzopth  9996
  Copyright terms: Public domain W3C validator