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Theorem nn0le2is012 9145
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 9086 . . . 4  |-  ( N  e.  NN0  ->  N  e.  ZZ )
2 2z 9094 . . . 4  |-  2  e.  ZZ
3 zleloe 9113 . . . 4  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <_  2  <->  ( N  <  2  \/  N  =  2 ) ) )
41, 2, 3sylancl 409 . . 3  |-  ( N  e.  NN0  ->  ( N  <_  2  <->  ( N  <  2  \/  N  =  2 ) ) )
5 zltlem1 9123 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
61, 2, 5sylancl 409 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
7 2m1e1 8850 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
87a1i 9 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
98breq2d 3941 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  ( 2  -  1 )  <->  N  <_  1 ) )
106, 9bitrd 187 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  1 ) )
11 1z 9092 . . . . . . . . 9  |-  1  e.  ZZ
12 zleloe 9113 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  ->  ( N  <_  1  <->  ( N  <  1  \/  N  =  1 ) ) )
131, 11, 12sylancl 409 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  1  <->  ( N  <  1  \/  N  =  1 ) ) )
14 nn0lt10b 9143 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  = 
0 ) )
15 3mix1 1150 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
1614, 15syl6bi 162 . . . . . . . . . . 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 1151 . . . . . . . . . . 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 705 . . . . . . . . 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 149 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <_  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2310, 22sylbid 149 . . . . . 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 1152 . . . . . 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 705 . . . 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 149 . 2  |-  ( N  e.  NN0  ->  ( N  <_  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
3029imp 123 1  |-  ( ( N  e.  NN0  /\  N  <_  2 )  -> 
( N  =  0  \/  N  =  1  \/  N  =  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   0cc0 7632   1c1 7633    < clt 7812    <_ cle 7813    - cmin 7945   2c2 8783   NN0cn0 8989   ZZcz 9066
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-2 8791  df-n0 8990  df-z 9067
This theorem is referenced by:  xnn0le2is012  9661
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