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Theorem nn0le2is012 9331
Description: A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
Assertion
Ref Expression
nn0le2is012  |-  ( ( N  e.  NN0  /\  N  <_  2 )  -> 
( N  =  0  \/  N  =  1  \/  N  =  2 ) )

Proof of Theorem nn0le2is012
StepHypRef Expression
1 nn0z 9269 . . . 4  |-  ( N  e.  NN0  ->  N  e.  ZZ )
2 2z 9277 . . . 4  |-  2  e.  ZZ
3 zleloe 9296 . . . 4  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <_  2  <->  ( N  <  2  \/  N  =  2 ) ) )
41, 2, 3sylancl 413 . . 3  |-  ( N  e.  NN0  ->  ( N  <_  2  <->  ( N  <  2  \/  N  =  2 ) ) )
5 zltlem1 9306 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
61, 2, 5sylancl 413 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
7 2m1e1 9033 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
87a1i 9 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
98breq2d 4014 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  ( 2  -  1 )  <->  N  <_  1 ) )
106, 9bitrd 188 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  1 ) )
11 1z 9275 . . . . . . . . 9  |-  1  e.  ZZ
12 zleloe 9296 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  ->  ( N  <_  1  <->  ( N  <  1  \/  N  =  1 ) ) )
131, 11, 12sylancl 413 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  1  <->  ( N  <  1  \/  N  =  1 ) ) )
14 nn0lt10b 9329 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  = 
0 ) )
15 3mix1 1166 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
1614, 15syl6bi 163 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  <  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
1716com12 30 . . . . . . . . . 10  |-  ( N  <  1  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
18 3mix2 1167 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
1918a1d 22 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2017, 19jaoi 716 . . . . . . . . 9  |-  ( ( N  <  1  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2120com12 30 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  <  1  \/  N  =  1 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2213, 21sylbid 150 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <_  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2310, 22sylbid 150 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  <  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2423com12 30 . . . . 5  |-  ( N  <  2  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
25 3mix3 1168 . . . . . 6  |-  ( N  =  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
2625a1d 22 . . . . 5  |-  ( N  =  2  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2724, 26jaoi 716 . . . 4  |-  ( ( N  <  2  \/  N  =  2 )  ->  ( N  e. 
NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2827com12 30 . . 3  |-  ( N  e.  NN0  ->  ( ( N  <  2  \/  N  =  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
294, 28sylbid 150 . 2  |-  ( N  e.  NN0  ->  ( N  <_  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
3029imp 124 1  |-  ( ( N  e.  NN0  /\  N  <_  2 )  -> 
( N  =  0  \/  N  =  1  \/  N  =  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    \/ w3o 977    = wceq 1353    e. wcel 2148   class class class wbr 4002  (class class class)co 5872   0cc0 7808   1c1 7809    < clt 7988    <_ cle 7989    - cmin 8124   2c2 8966   NN0cn0 9172   ZZcz 9249
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-inn 8916  df-2 8974  df-n0 9173  df-z 9250
This theorem is referenced by:  xnn0le2is012  9862
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