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Theorem nn0n0n1ge2b 9331
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 9322 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
213expib 1206 . 2  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N
) )
3 nn0z 9272 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 0z 9263 . . . . . 6  |-  0  e.  ZZ
5 zdceq 9327 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
63, 4, 5sylancl 413 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  0
)
76dcned 2353 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
8 1z 9278 . . . . . 6  |-  1  e.  ZZ
9 zdceq 9327 . . . . . 6  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  -> DECID  N  =  1 )
103, 8, 9sylancl 413 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  1
)
1110dcned 2353 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  1
)
12 dcan2 934 . . . 4  |-  (DECID  N  =/=  0  ->  (DECID  N  =/=  1  -> DECID 
( N  =/=  0  /\  N  =/=  1
) ) )
137, 11, 12sylc 62 . . 3  |-  ( N  e.  NN0  -> DECID  ( N  =/=  0  /\  N  =/=  1
) )
14 ianordc 899 . . . . . 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 2352 . . . . . . 7  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
176, 16syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  0  <->  N  =  0 ) )
18 nnedc 2352 . . . . . . 7  |-  (DECID  N  =  1  ->  ( -.  N  =/=  1  <->  N  = 
1 ) )
1910, 18syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  1  <->  N  =  1 ) )
2017, 19orbi12d 793 . . . . 5  |-  ( N  e.  NN0  ->  ( ( -.  N  =/=  0  \/  -.  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
2115, 20bitrd 188 . . . 4  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
22 2pos 9009 . . . . . . . . . 10  |-  0  <  2
23 breq1 4006 . . . . . . . . . 10  |-  ( N  =  0  ->  ( N  <  2  <->  0  <  2 ) )
2422, 23mpbiri 168 . . . . . . . . 9  |-  ( N  =  0  ->  N  <  2 )
2524a1d 22 . . . . . . . 8  |-  ( N  =  0  ->  ( N  e.  NN0  ->  N  <  2 ) )
26 1lt2 9087 . . . . . . . . . 10  |-  1  <  2
27 breq1 4006 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  <  2  <->  1  <  2 ) )
2826, 27mpbiri 168 . . . . . . . . 9  |-  ( N  =  1  ->  N  <  2 )
2928a1d 22 . . . . . . . 8  |-  ( N  =  1  ->  ( N  e.  NN0  ->  N  <  2 ) )
3025, 29jaoi 716 . . . . . . 7  |-  ( ( N  =  0  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  N  <  2
) )
3130impcom 125 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  N  <  2 )
32 2z 9280 . . . . . . . 8  |-  2  e.  ZZ
33 zltnle 9298 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  -.  2  <_  N )
)
343, 32, 33sylancl 413 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  -.  2  <_  N ) )
3534adantr 276 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  ( N  <  2  <->  -.  2  <_  N ) )
3631, 35mpbid 147 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  -.  2  <_  N )
3736ex 115 . . . 4  |-  ( N  e.  NN0  ->  ( ( N  =  0  \/  N  =  1 )  ->  -.  2  <_  N ) )
3821, 37sylbid 150 . . 3  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  ->  -.  2  <_  N ) )
39 condc 853 . . 3  |-  (DECID  ( N  =/=  0  /\  N  =/=  1 )  ->  (
( -.  ( N  =/=  0  /\  N  =/=  1 )  ->  -.  2  <_  N )  -> 
( 2  <_  N  ->  ( N  =/=  0  /\  N  =/=  1
) ) ) )
4013, 38, 39sylc 62 . 2  |-  ( N  e.  NN0  ->  ( 2  <_  N  ->  ( N  =/=  0  /\  N  =/=  1 ) ) )
412, 40impbid 129 1  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  <->  2  <_  N )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148    =/= wne 2347   class class class wbr 4003   0cc0 7810   1c1 7811    < clt 7991    <_ cle 7992   2c2 8969   NN0cn0 9175   ZZcz 9252
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  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-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-2 8977  df-n0 9176  df-z 9253
This theorem is referenced by: (None)
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