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Theorem nn0n0n1ge2b 9084
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 9075 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
213expib 1167 . 2  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N
) )
3 nn0z 9028 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 0z 9019 . . . . . 6  |-  0  e.  ZZ
5 zdceq 9080 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
63, 4, 5sylancl 407 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  0
)
76dcned 2289 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
8 1z 9034 . . . . . 6  |-  1  e.  ZZ
9 zdceq 9080 . . . . . 6  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  -> DECID  N  =  1 )
103, 8, 9sylancl 407 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  1
)
1110dcned 2289 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  1
)
12 dcan 901 . . . 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 867 . . . . . 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 2288 . . . . . . 7  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
176, 16syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  0  <->  N  =  0 ) )
18 nnedc 2288 . . . . . . 7  |-  (DECID  N  =  1  ->  ( -.  N  =/=  1  <->  N  = 
1 ) )
1910, 18syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  1  <->  N  =  1 ) )
2017, 19orbi12d 765 . . . . 5  |-  ( N  e.  NN0  ->  ( ( -.  N  =/=  0  \/  -.  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
2115, 20bitrd 187 . . . 4  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  <->  ( N  =  0  \/  N  =  1 ) ) )
22 2pos 8771 . . . . . . . . . 10  |-  0  <  2
23 breq1 3900 . . . . . . . . . 10  |-  ( N  =  0  ->  ( N  <  2  <->  0  <  2 ) )
2422, 23mpbiri 167 . . . . . . . . 9  |-  ( N  =  0  ->  N  <  2 )
2524a1d 22 . . . . . . . 8  |-  ( N  =  0  ->  ( N  e.  NN0  ->  N  <  2 ) )
26 1lt2 8843 . . . . . . . . . 10  |-  1  <  2
27 breq1 3900 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  <  2  <->  1  <  2 ) )
2826, 27mpbiri 167 . . . . . . . . 9  |-  ( N  =  1  ->  N  <  2 )
2928a1d 22 . . . . . . . 8  |-  ( N  =  1  ->  ( N  e.  NN0  ->  N  <  2 ) )
3025, 29jaoi 688 . . . . . . 7  |-  ( ( N  =  0  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  N  <  2
) )
3130impcom 124 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  N  <  2 )
32 2z 9036 . . . . . . . 8  |-  2  e.  ZZ
33 zltnle 9054 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  -.  2  <_  N )
)
343, 32, 33sylancl 407 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  -.  2  <_  N ) )
3534adantr 272 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  ( N  <  2  <->  -.  2  <_  N ) )
3631, 35mpbid 146 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  -.  2  <_  N )
3736ex 114 . . . 4  |-  ( N  e.  NN0  ->  ( ( N  =  0  \/  N  =  1 )  ->  -.  2  <_  N ) )
3821, 37sylbid 149 . . 3  |-  ( N  e.  NN0  ->  ( -.  ( N  =/=  0  /\  N  =/=  1
)  ->  -.  2  <_  N ) )
39 condc 821 . . 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 128 1  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  <->  2  <_  N )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283   class class class wbr 3897   0cc0 7584   1c1 7585    < clt 7764    <_ cle 7765   2c2 8731   NN0cn0 8931   ZZcz 9008
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-2 8739  df-n0 8932  df-z 9009
This theorem is referenced by: (None)
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