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Theorem nn0n0n1ge2b 8544
Description: A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
Assertion
Ref Expression
nn0n0n1ge2b  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  <->  2  <_  N )
)

Proof of Theorem nn0n0n1ge2b
StepHypRef Expression
1 nn0n0n1ge2 8535 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
213expib 1142 . 2  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N
) )
3 nn0z 8488 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 0z 8479 . . . . . 6  |-  0  e.  ZZ
5 zdceq 8540 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
63, 4, 5sylancl 404 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  0
)
76dcned 2255 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
8 1z 8494 . . . . . 6  |-  1  e.  ZZ
9 zdceq 8540 . . . . . 6  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  -> DECID  N  =  1 )
103, 8, 9sylancl 404 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  1
)
1110dcned 2255 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  1
)
12 dcan 876 . . . 4  |-  (DECID  N  =/=  0  ->  (DECID  N  =/=  1  -> DECID 
( N  =/=  0  /\  N  =/=  1
) ) )
137, 11, 12sylc 61 . . 3  |-  ( N  e.  NN0  -> DECID  ( N  =/=  0  /\  N  =/=  1
) )
14 ianordc 833 . . . . . 6  |-  (DECID  N  =/=  0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  <->  ( -.  N  =/=  0  \/  -.  N  =/=  1 ) ) )
157, 14syl 14 . . . . 5  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  <->  ( -.  N  =/=  0  \/  -.  N  =/=  1 ) ) )
16 nnedc 2254 . . . . . . 7  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
176, 16syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  0  <->  N  =  0 ) )
18 nnedc 2254 . . . . . . 7  |-  (DECID  N  =  1  ->  ( -.  N  =/=  1  <->  N  = 
1 ) )
1910, 18syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  1  <->  N  =  1 ) )
2017, 19orbi12d 740 . . . . 5  |-  ( N  e.  NN0  ->  ( ( -.  N  =/=  0  \/  -.  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
2115, 20bitrd 186 . . . 4  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
22 2pos 8233 . . . . . . . . . 10  |-  0  <  2
23 breq1 3809 . . . . . . . . . 10  |-  ( N  =  0  ->  ( N  <  2  <->  0  <  2 ) )
2422, 23mpbiri 166 . . . . . . . . 9  |-  ( N  =  0  ->  N  <  2 )
2524a1d 22 . . . . . . . 8  |-  ( N  =  0  ->  ( N  e.  NN0  ->  N  <  2 ) )
26 1lt2 8304 . . . . . . . . . 10  |-  1  <  2
27 breq1 3809 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  <  2  <->  1  <  2 ) )
2826, 27mpbiri 166 . . . . . . . . 9  |-  ( N  =  1  ->  N  <  2 )
2928a1d 22 . . . . . . . 8  |-  ( N  =  1  ->  ( N  e.  NN0  ->  N  <  2 ) )
3025, 29jaoi 669 . . . . . . 7  |-  ( ( N  =  0  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  N  <  2
) )
3130impcom 123 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  N  <  2 )
32 2z 8496 . . . . . . . 8  |-  2  e.  ZZ
33 zltnle 8514 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  -.  2  <_  N )
)
343, 32, 33sylancl 404 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  -.  2  <_  N ) )
3534adantr 270 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  ( N  <  2  <->  -.  2  <_  N ) )
3631, 35mpbid 145 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  -.  2  <_  N )
3736ex 113 . . . 4  |-  ( N  e.  NN0  ->  ( ( N  =  0  \/  N  =  1 )  ->  -.  2  <_  N ) )
3821, 37sylbid 148 . . 3  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  ->  -.  2  <_  N ) )
39 condc 783 . . 3  |-  (DECID  ( N  =/=  0  /\  N  =/=  1 )  ->  (
( -.  ( N  =/=  0  /\  N  =/=  1 )  ->  -.  2  <_  N )  -> 
( 2  <_  N  ->  ( N  =/=  0  /\  N  =/=  1
) ) ) )
4013, 38, 39sylc 61 . 2  |-  ( N  e.  NN0  ->  ( 2  <_  N  ->  ( N  =/=  0  /\  N  =/=  1 ) ) )
412, 40impbid 127 1  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  <->  2  <_  N )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3806   0cc0 7079   1c1 7080    < clt 7251    <_ cle 7252   2c2 8192   NN0cn0 8391   ZZcz 8468
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 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-2 8201  df-n0 8392  df-z 8469
This theorem is referenced by: (None)
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