ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn0n0n1ge2 Unicode version

Theorem nn0n0n1ge2 8815
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 8681 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
2 1cnd 7502 . . . . . 6  |-  ( N  e.  NN0  ->  1  e.  CC )
31, 2, 2subsub4d 7822 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  -  1 )  =  ( N  -  (
1  +  1 ) ) )
4 1p1e2 8537 . . . . . 6  |-  ( 1  +  1 )  =  2
54oveq2i 5663 . . . . 5  |-  ( N  -  ( 1  +  1 ) )  =  ( N  -  2 )
63, 5syl6req 2137 . . . 4  |-  ( N  e.  NN0  ->  ( N  -  2 )  =  ( ( N  - 
1 )  -  1 ) )
763ad2ant1 964 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  =  ( ( N  -  1 )  - 
1 ) )
8 3simpa 940 . . . . . . 7  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  e.  NN0  /\  N  =/=  0 ) )
9 elnnne0 8685 . . . . . . 7  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
108, 9sylibr 132 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  N  e.  NN )
11 nnm1nn0 8712 . . . . . 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 7801 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  <->  N  = 
1 ) )
1413biimpd 142 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  =  0  ->  N  =  1 ) )
1514necon3d 2299 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  =/=  1  ->  ( N  -  1 )  =/=  0 ) )
1615imp 122 . . . . . 6  |-  ( ( N  e.  NN0  /\  N  =/=  1 )  -> 
( N  -  1 )  =/=  0 )
17163adant2 962 . . . . 5  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  =/=  0 )
18 elnnne0 8685 . . . . 5  |-  ( ( N  -  1 )  e.  NN  <->  ( ( N  -  1 )  e.  NN0  /\  ( N  -  1 )  =/=  0 ) )
1912, 17, 18sylanbrc 408 . . . 4  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  1 )  e.  NN )
20 nnm1nn0 8712 . . . 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 2164 . 2  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  ( N  -  2 )  e.  NN0 )
23 2nn0 8688 . . . . 5  |-  2  e.  NN0
2423jctl 307 . . . 4  |-  ( N  e.  NN0  ->  ( 2  e.  NN0  /\  N  e. 
NN0 ) )
25243ad2ant1 964 . . 3  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  (
2  e.  NN0  /\  N  e.  NN0 ) )
26 nn0sub 8814 . . 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 165 1  |-  ( ( N  e.  NN0  /\  N  =/=  0  /\  N  =/=  1 )  ->  2  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845  (class class class)co 5652   0cc0 7348   1c1 7349    + caddc 7351    <_ cle 7521    - cmin 7651   NNcn 8420   2c2 8471   NN0cn0 8671
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 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  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 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-2 8479  df-n0 8672  df-z 8749
This theorem is referenced by:  nn0n0n1ge2b  8824
  Copyright terms: Public domain W3C validator