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Theorem nn0n0n1ge2b 8796
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 8787 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
213expib 1146 . 2  |-  ( N  e.  NN0  ->  ( ( N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N
) )
3 nn0z 8740 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 0z 8731 . . . . . 6  |-  0  e.  ZZ
5 zdceq 8792 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
63, 4, 5sylancl 404 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  0
)
76dcned 2261 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  0
)
8 1z 8746 . . . . . 6  |-  1  e.  ZZ
9 zdceq 8792 . . . . . 6  |-  ( ( N  e.  ZZ  /\  1  e.  ZZ )  -> DECID  N  =  1 )
103, 8, 9sylancl 404 . . . . 5  |-  ( N  e.  NN0  -> DECID  N  =  1
)
1110dcned 2261 . . . 4  |-  ( N  e.  NN0  -> DECID  N  =/=  1
)
12 dcan 880 . . . 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 837 . . . . . 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 2260 . . . . . . 7  |-  (DECID  N  =  0  ->  ( -.  N  =/=  0  <->  N  = 
0 ) )
176, 16syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  0  <->  N  =  0 ) )
18 nnedc 2260 . . . . . . 7  |-  (DECID  N  =  1  ->  ( -.  N  =/=  1  <->  N  = 
1 ) )
1910, 18syl 14 . . . . . 6  |-  ( N  e.  NN0  ->  ( -.  N  =/=  1  <->  N  =  1 ) )
2017, 19orbi12d 742 . . . . 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 8484 . . . . . . . . . 10  |-  0  <  2
23 breq1 3840 . . . . . . . . . 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 8555 . . . . . . . . . 10  |-  1  <  2
27 breq1 3840 . . . . . . . . . 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 671 . . . . . . 7  |-  ( ( N  =  0  \/  N  =  1 )  ->  ( N  e. 
NN0  ->  N  <  2
) )
3130impcom 123 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( N  =  0  \/  N  =  1
) )  ->  N  <  2 )
32 2z 8748 . . . . . . . 8  |-  2  e.  ZZ
33 zltnle 8766 . . . . . . . 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 787 . . 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 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3837   0cc0 7329   1c1 7330    < clt 7501    <_ cle 7502   2c2 8444   NN0cn0 8643   ZZcz 8720
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721
This theorem is referenced by: (None)
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