Theorem nn0n0n1ge2 8418

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 8298	. . . . . 6 |- ( N e. NN0 -> N e. CC )
2		1cnd 7135	. . . . . 6 |- ( N e. NN0 -> 1 e. CC )
3	1, 2, 2	subsub4d 7450	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
4		1p1e2 8155	. . . . . 6 |- ( 1 + 1 ) = 2
5	4	oveq2i 5543	. . . . 5 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
6	3, 5	syl6req 2130	. . . 4 |- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
7	6	3ad2ant1 959	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
8		3simpa 935	. . . . . . 7 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) )
9		elnnne0 8302	. . . . . . 7 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
10	8, 9	sylibr 132	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> N e. NN )
11		nnm1nn0 8329	. . . . . 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 7429	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) = 0 <-> N = 1 ) )
14	13	biimpd 142	. . . . . . . 8 |- ( N e. NN0 -> ( ( N - 1 ) = 0 -> N = 1 ) )
15	14	necon3d 2289	. . . . . . 7 |- ( N e. NN0 -> ( N =/= 1 -> ( N - 1 ) =/= 0 ) )
16	15	imp 122	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
17	16	3adant2 957	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
18		elnnne0 8302	. . . . 5 |- ( ( N - 1 ) e. NN <-> ( ( N - 1 ) e. NN0 /\ ( N - 1 ) =/= 0 ) )
19	12, 17, 18	sylanbrc 408	. . . 4 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN )
20		nnm1nn0 8329	. . . 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 2155	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 )
23		2nn0 8305	. . . . 5 |- 2 e. NN0
24	23	jctl 307	. . . 4 |- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) )
25	24	3ad2ant1 959	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) )
26		nn0sub 8417	. . 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 165	1 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   class class class wbr 3785  (class class class)co 5532   0cc0 6981   1c1 6982   + caddc 6984   <_ cle 7154   - cmin 7279   NNcn 8039   2c2 8089   NN0cn0 8288

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352

This theorem is referenced by:  nn0n0n1ge2b  8427