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Theorem nn0n0n1ge2b 9098
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 9089 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
213expib 1169 . 2  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N
) )
3 nn0z 9042 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 0z 9033 . . . . . 6  |-  0  e.  ZZ
5 zdceq 9094 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
63, 4, 5sylancl 409 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  0
)
76dcned 2291 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
8 1z 9048 . . . . . 6  |-  1  e.  ZZ
9 zdceq 9094 . . . . . 6  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  -> DECID  N  =  1 )
103, 8, 9sylancl 409 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  1
)
1110dcned 2291 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  1
)
12 dcan 903 . . . 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 869 . . . . . 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 2290 . . . . . . 7  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
176, 16syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  0  <->  N  =  0 ) )
18 nnedc 2290 . . . . . . 7  |-  (DECID  N  =  1  ->  ( -.  N  =/=  1  <->  N  = 
1 ) )
1910, 18syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  1  <->  N  =  1 ) )
2017, 19orbi12d 767 . . . . 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 8779 . . . . . . . . . 10  |-  0  <  2
23 breq1 3902 . . . . . . . . . 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 8857 . . . . . . . . . 10  |-  1  <  2
27 breq1 3902 . . . . . . . . . 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 690 . . . . . . 7  |-  ( ( N  =  0  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  N  <  2
) )
3130impcom 124 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  N  <  2 )
32 2z 9050 . . . . . . . 8  |-  2  e.  ZZ
33 zltnle 9068 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  2  e.  ZZ )  ->  ( N  <  2  <->  -.  2  <_  N )
)
343, 32, 33sylancl 409 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  <  2  <->  -.  2  <_  N ) )
3534adantr 274 . . . . . 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 823 . . 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 682  DECID wdc 804    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899   0cc0 7588   1c1 7589    < clt 7768    <_ cle 7769   2c2 8739   NN0cn0 8945   ZZcz 9022
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023
This theorem is referenced by: (None)
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