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Theorem nn0le2is012 30906
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 nn0re 10689 . . . 4  |-  ( N  e.  NN0  ->  N  e.  RR )
2 2re 10492 . . . . 5  |-  2  e.  RR
32a1i 11 . . . 4  |-  ( N  e.  NN0  ->  2  e.  RR )
41, 3leloed 9618 . . 3  |-  ( N  e.  NN0  ->  ( N  <_  2  <->  ( N  <  2  \/  N  =  2 ) ) )
5 nn0z 10770 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
6 2z 10779 . . . . . . . . 9  |-  2  e.  ZZ
7 zltlem1 10798 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
85, 6, 7sylancl 662 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  ( 2  -  1 ) ) )
9 2m1e1 10537 . . . . . . . . . 10  |-  ( 2  -  1 )  =  1
109a1i 11 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
1110breq2d 4402 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  ( 2  -  1 )  <->  N  <_  1 ) )
128, 11bitrd 253 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  N  <_  1 ) )
13 1red 9502 . . . . . . . . 9  |-  ( N  e.  NN0  ->  1  e.  RR )
141, 13leloed 9618 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  <_  1  <->  ( N  <  1  \/  N  =  1 ) ) )
15 nn0lt10b 10807 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  = 
0 ) )
16 3mix1 1157 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
1715, 16syl6bi 228 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  <  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
1817com12 31 . . . . . . . . . 10  |-  ( N  <  1  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
19 3mix2 1158 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
2019a1d 25 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2118, 20jaoi 379 . . . . . . . . 9  |-  ( ( N  <  1  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2221com12 31 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  <  1  \/  N  =  1 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2314, 22sylbid 215 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <_  1  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2412, 23sylbid 215 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  <  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2524com12 31 . . . . 5  |-  ( N  <  2  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
26 3mix3 1159 . . . . . 6  |-  ( N  =  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
2726a1d 25 . . . . 5  |-  ( N  =  2  ->  ( N  e.  NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2825, 27jaoi 379 . . . 4  |-  ( ( N  <  2  \/  N  =  2 )  ->  ( N  e. 
NN0  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
2928com12 31 . . 3  |-  ( N  e.  NN0  ->  ( ( N  <  2  \/  N  =  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
304, 29sylbid 215 . 2  |-  ( N  e.  NN0  ->  ( N  <_  2  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) ) )
3130imp 429 1  |-  ( ( N  e.  NN0  /\  N  <_  2 )  -> 
( N  =  0  \/  N  =  1  \/  N  =  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758   class class class wbr 4390  (class class class)co 6190   RRcr 9382   0cc0 9383   1c1 9384    < clt 9519    <_ cle 9520    - cmin 9696   2c2 10472   NN0cn0 10680   ZZcz 10747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748
This theorem is referenced by:  exple2lt6  30907
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