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Theorem uz11 8711
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 8703 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
2 eleq2 2143 . . . . . 6  |-  ( (
ZZ>= `  M )  =  ( ZZ>= `  N )  ->  ( M  e.  (
ZZ>= `  M )  <->  M  e.  ( ZZ>= `  N )
) )
3 eluzel2 8694 . . . . . 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 8703 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
7 eleq2 2143 . . . . . . . . . . 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 8701 . . . . . . . . . 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 8701 . . . . . . . . . 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 8425 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
19 zre 8425 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  RR )
20 letri3 7248 . . . . . . 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 5203 . 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 1285    e. wcel 1434   class class class wbr 3787   ` cfv 4926   RRcr 7031    <_ cle 7205   ZZcz 8421   ZZ>=cuz 8689
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-pre-ltirr 7139  ax-pre-apti 7142
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-ov 5540  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-neg 7338  df-z 8422  df-uz 8690
This theorem is referenced by:  fzopth  9144
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