ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uz11 Unicode version

Theorem uz11 9482
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 9474 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
2 eleq2 2228 . . . . . 6  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  (
ZZ>= `  M )  <->  M  e.  ( ZZ>= `  N )
) )
3 eluzel2 9465 . . . . . 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 9474 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
7 eleq2 2228 . . . . . . . . . . 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 9472 . . . . . . . . . 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 9472 . . . . . . . . . 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 9189 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
19 zre 9189 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  RR )
20 letri3 7973 . . . . . . 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 5483 . 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 1342    e. wcel 2135   class class class wbr 3979   ` cfv 5185   RRcr 7746    <_ cle 7928   ZZcz 9185   ZZ>=cuz 9460
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-pre-ltirr 7859  ax-pre-apti 7862
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-mpt 4042  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-fv 5193  df-ov 5842  df-pnf 7929  df-mnf 7930  df-xr 7931  df-ltxr 7932  df-le 7933  df-neg 8066  df-z 9186  df-uz 9461
This theorem is referenced by:  fzopth  9990
  Copyright terms: Public domain W3C validator