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Theorem nn0n0n1ge2 9079
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 8945 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
2 1cnd 7750 . . . . . 6  |-  ( N  e.  NN0  ->  1  e.  CC )
31, 2, 2subsub4d 8072 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
4 1p1e2 8801 . . . . . 6  |-  ( 1  +  1 )  =  2
54oveq2i 5753 . . . . 5  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
63, 5syl6req 2167 . . . 4  |-  ( N  e.  NN0  ->  ( N  -  2 )  =  ( ( N  - 
1 )  -  1 ) )
763ad2ant1 987 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  =  ( ( N  -  1 )  - 
1 ) )
8 3simpa 963 . . . . . . 7  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  e.  NN0  /\  N  =/=  0 ) )
9 elnnne0 8949 . . . . . . 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 8976 . . . . . 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 8051 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  <->  N  = 
1 ) )
1413biimpd 143 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  ->  N  =  1 ) )
1514necon3d 2329 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  =/=  1  ->  ( N  -  1 )  =/=  0 ) )
1615imp 123 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  1 )  -> 
( N  -  1 )  =/=  0 )
17163adant2 985 . . . . 5  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
18 elnnne0 8949 . . . . 5  |-  ( ( N  -  1 )  e.  NN  <->  ( ( N  -  1 )  e.  NN0  /\  ( N  -  1 )  =/=  0 ) )
1912, 17, 18sylanbrc 413 . . . 4  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  e.  NN )
20 nnm1nn0 8976 . . . 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 2194 . 2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  e.  NN0 )
23 2nn0 8952 . . . . 5  |-  2  e.  NN0
2423jctl 312 . . . 4  |-  ( N  e.  NN0  ->  ( 2  e.  NN0  /\  N  e. 
NN0 ) )
25243ad2ant1 987 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
2  e.  NN0  /\  N  e.  NN0 ) )
26 nn0sub 9078 . . 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 947    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899  (class class class)co 5742   0cc0 7588   1c1 7589    + caddc 7591    <_ cle 7769    - cmin 7901   NNcn 8684   2c2 8735   NN0cn0 8935
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-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 8685  df-2 8743  df-n0 8936  df-z 9013
This theorem is referenced by:  nn0n0n1ge2b  9088
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