Theorem nn0n0n1ge2 9261

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

Step	Hyp	Ref	Expression
1		nn0cn 9124	. . . . . 6 |- ( N e. NN0 -> N e. CC )
2		1cnd 7915	. . . . . 6 |- ( N e. NN0 -> 1 e. CC )
3	1, 2, 2	subsub4d 8240	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
4		1p1e2 8974	. . . . . 6 |- ( 1 + 1 ) = 2
5	4	oveq2i 5853	. . . . 5 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
6	3, 5	eqtr2di 2216	. . . 4 |- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
7	6	3ad2ant1 1008	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
8		3simpa 984	. . . . . . 7 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) )
9		elnnne0 9128	. . . . . . 7 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
10	8, 9	sylibr 133	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> N e. NN )
11		nnm1nn0 9155	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
12	10, 11	syl 14	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN0 )
13	1, 2	subeq0ad 8219	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) = 0 <-> N = 1 ) )
14	13	biimpd 143	. . . . . . . 8 |- ( N e. NN0 -> ( ( N - 1 ) = 0 -> N = 1 ) )
15	14	necon3d 2380	. . . . . . 7 |- ( N e. NN0 -> ( N =/= 1 -> ( N - 1 ) =/= 0 ) )
16	15	imp 123	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
17	16	3adant2 1006	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
18		elnnne0 9128	. . . . 5 |- ( ( N - 1 ) e. NN <-> ( ( N - 1 ) e. NN0 /\ ( N - 1 ) =/= 0 ) )
19	12, 17, 18	sylanbrc 414	. . . 4 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN )
20		nnm1nn0 9155	. . . 4 |- ( ( N - 1 ) e. NN -> ( ( N - 1 ) - 1 ) e. NN0 )
21	19, 20	syl 14	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( ( N - 1 ) - 1 ) e. NN0 )
22	7, 21	eqeltrd 2243	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 )
23		2nn0 9131	. . . . 5 |- 2 e. NN0
24	23	jctl 312	. . . 4 |- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) )
25	24	3ad2ant1 1008	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) )
26		nn0sub 9257	. . 3 |- ( ( 2 e. NN0 /\ N e. NN0 ) -> ( 2 <_ N <-> ( N - 2 ) e. NN0 ) )
27	25, 26	syl 14	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 <_ N <-> ( N - 2 ) e. NN0 ) )
28	22, 27	mpbird 166	1 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   0cc0 7753   1c1 7754   + caddc 7756   <_ cle 7934   - cmin 8069   NNcn 8857   2c2 8908   NN0cn0 9114

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192

This theorem is referenced by:  nn0n0n1ge2b  9270