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Theorem nn0n0n1ge2 9282
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 9145 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
2 1cnd 7936 . . . . . 6  |-  ( N  e.  NN0  ->  1  e.  CC )
31, 2, 2subsub4d 8261 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
4 1p1e2 8995 . . . . . 6  |-  ( 1  +  1 )  =  2
54oveq2i 5864 . . . . 5  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
63, 5eqtr2di 2220 . . . 4  |-  ( N  e.  NN0  ->  ( N  -  2 )  =  ( ( N  - 
1 )  -  1 ) )
763ad2ant1 1013 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  =  ( ( N  -  1 )  - 
1 ) )
8 3simpa 989 . . . . . . 7  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  e.  NN0  /\  N  =/=  0 ) )
9 elnnne0 9149 . . . . . . 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 9176 . . . . . 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 8240 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  <->  N  = 
1 ) )
1413biimpd 143 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  ->  N  =  1 ) )
1514necon3d 2384 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  =/=  1  ->  ( N  -  1 )  =/=  0 ) )
1615imp 123 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  1 )  -> 
( N  -  1 )  =/=  0 )
17163adant2 1011 . . . . 5  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
18 elnnne0 9149 . . . . 5  |-  ( ( N  -  1 )  e.  NN  <->  ( ( N  -  1 )  e.  NN0  /\  ( N  -  1 )  =/=  0 ) )
1912, 17, 18sylanbrc 415 . . . 4  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  e.  NN )
20 nnm1nn0 9176 . . . 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 2247 . 2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  e.  NN0 )
23 2nn0 9152 . . . . 5  |-  2  e.  NN0
2423jctl 312 . . . 4  |-  ( N  e.  NN0  ->  ( 2  e.  NN0  /\  N  e. 
NN0 ) )
25243ad2ant1 1013 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
2  e.  NN0  /\  N  e.  NN0 ) )
26 nn0sub 9278 . . 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 973    = wceq 1348    e. wcel 2141    =/= wne 2340   class class class wbr 3989  (class class class)co 5853   0cc0 7774   1c1 7775    + caddc 7777    <_ cle 7955    - cmin 8090   NNcn 8878   2c2 8929   NN0cn0 9135
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213
This theorem is referenced by:  nn0n0n1ge2b  9291
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