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Theorem uz11 9041
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 9033 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
2 eleq2 2151 . . . . . 6  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  (
ZZ>= `  M )  <->  M  e.  ( ZZ>= `  N )
) )
3 eluzel2 9024 . . . . . 6  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
42, 3syl6bi 161 . . . . 5  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  (
ZZ>= `  M )  ->  N  e.  ZZ )
)
51, 4mpan9 275 . . . 4  |-  ( ( M  e.  ZZ  /\  ( ZZ>= `  M )  =  ( ZZ>= `  N
) )  ->  N  e.  ZZ )
6 uzid 9033 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
7 eleq2 2151 . . . . . . . . . . 11  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( N  e.  (
ZZ>= `  M )  <->  N  e.  ( ZZ>= `  N )
) )
86, 7syl5ibr 154 . . . . . . . . . 10  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  M ) ) )
9 eluzle 9031 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
108, 9syl6 33 . . . . . . . . 9  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( N  e.  ZZ  ->  M  <_  N )
)
111, 2syl5ib 152 . . . . . . . . . 10  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  N ) ) )
12 eluzle 9031 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
1311, 12syl6 33 . . . . . . . . 9  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  ZZ  ->  N  <_  M )
)
1410, 13anim12d 328 . . . . . . . 8  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( M  <_  N  /\  N  <_  M ) ) )
1514impl 372 . . . . . . 7  |-  ( ( ( ( ZZ>= `  M
)  =  ( ZZ>= `  N )  /\  N  e.  ZZ )  /\  M  e.  ZZ )  ->  ( M  <_  N  /\  N  <_  M ) )
1615ancoms 264 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( ( ZZ>= `  M
)  =  ( ZZ>= `  N )  /\  N  e.  ZZ ) )  -> 
( M  <_  N  /\  N  <_  M ) )
1716anassrs 392 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  ( ZZ>= `  M )  =  ( ZZ>= `  N
) )  /\  N  e.  ZZ )  ->  ( M  <_  N  /\  N  <_  M ) )
18 zre 8754 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
19 zre 8754 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  RR )
20 letri3 7566 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2118, 19, 20syl2an 283 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2221adantlr 461 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  ( ZZ>= `  M )  =  ( ZZ>= `  N
) )  /\  N  e.  ZZ )  ->  ( M  =  N  <->  ( M  <_  N  /\  N  <_  M ) ) )
2317, 22mpbird 165 . . . 4  |-  ( ( ( M  e.  ZZ  /\  ( ZZ>= `  M )  =  ( ZZ>= `  N
) )  /\  N  e.  ZZ )  ->  M  =  N )
245, 23mpdan 412 . . 3  |-  ( ( M  e.  ZZ  /\  ( ZZ>= `  M )  =  ( ZZ>= `  N
) )  ->  M  =  N )
2524ex 113 . 2  |-  ( M  e.  ZZ  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  N
)  ->  M  =  N ) )
26 fveq2 5305 . 2  |-  ( M  =  N  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  N )
)
2725, 26impbid1 140 1  |-  ( M  e.  ZZ  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  N
)  <->  M  =  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845   ` cfv 5015   RRcr 7349    <_ cle 7523   ZZcz 8750   ZZ>=cuz 9019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-pre-ltirr 7457  ax-pre-apti 7460
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-ov 5655  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-neg 7656  df-z 8751  df-uz 9020
This theorem is referenced by:  fzopth  9475
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