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Theorem nn0n0n1ge2 9234
Description: A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
nn0n0n1ge2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )

Proof of Theorem nn0n0n1ge2
StepHypRef Expression
1 nn0cn 9100 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
2 1cnd 7894 . . . . . 6  |-  ( N  e.  NN0  ->  1  e.  CC )
31, 2, 2subsub4d 8217 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
4 1p1e2 8950 . . . . . 6  |-  ( 1  +  1 )  =  2
54oveq2i 5835 . . . . 5  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
63, 5eqtr2di 2207 . . . 4  |-  ( N  e.  NN0  ->  ( N  -  2 )  =  ( ( N  - 
1 )  -  1 ) )
763ad2ant1 1003 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  =  ( ( N  -  1 )  - 
1 ) )
8 3simpa 979 . . . . . . 7  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  e.  NN0  /\  N  =/=  0 ) )
9 elnnne0 9104 . . . . . . 7  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
108, 9sylibr 133 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  N  e.  NN )
11 nnm1nn0 9131 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
1210, 11syl 14 . . . . 5  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  e.  NN0 )
131, 2subeq0ad 8196 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  <->  N  = 
1 ) )
1413biimpd 143 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  ->  N  =  1 ) )
1514necon3d 2371 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  =/=  1  ->  ( N  -  1 )  =/=  0 ) )
1615imp 123 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  1 )  -> 
( N  -  1 )  =/=  0 )
17163adant2 1001 . . . . 5  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
18 elnnne0 9104 . . . . 5  |-  ( ( N  -  1 )  e.  NN  <->  ( ( N  -  1 )  e.  NN0  /\  ( N  -  1 )  =/=  0 ) )
1912, 17, 18sylanbrc 414 . . . 4  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  e.  NN )
20 nnm1nn0 9131 . . . 4  |-  ( ( N  -  1 )  e.  NN  ->  (
( N  -  1 )  -  1 )  e.  NN0 )
2119, 20syl 14 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
( N  -  1 )  -  1 )  e.  NN0 )
227, 21eqeltrd 2234 . 2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  e.  NN0 )
23 2nn0 9107 . . . . 5  |-  2  e.  NN0
2423jctl 312 . . . 4  |-  ( N  e.  NN0  ->  ( 2  e.  NN0  /\  N  e. 
NN0 ) )
25243ad2ant1 1003 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
2  e.  NN0  /\  N  e.  NN0 ) )
26 nn0sub 9233 . . 3  |-  ( ( 2  e.  NN0  /\  N  e.  NN0 )  -> 
( 2  <_  N  <->  ( N  -  2 )  e.  NN0 ) )
2725, 26syl 14 . 2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
2  <_  N  <->  ( N  -  2 )  e. 
NN0 ) )
2822, 27mpbird 166 1  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3965  (class class class)co 5824   0cc0 7732   1c1 7733    + caddc 7735    <_ cle 7913    - cmin 8046   NNcn 8833   2c2 8884   NN0cn0 9090
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168
This theorem is referenced by:  nn0n0n1ge2b  9243
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