Theorem nn0n0n1ge2 9650

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 9508	. . . . . 6 |- ( N e. NN0 -> N e. CC )
2		1cnd 8292	. . . . . 6 |- ( N e. NN0 -> 1 e. CC )
3	1, 2, 2	subsub4d 8617	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
4		1p1e2 9356	. . . . . 6 |- ( 1 + 1 ) = 2
5	4	oveq2i 6063	. . . . 5 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
6	3, 5	eqtr2di 2284	. . . 4 |- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
7	6	3ad2ant1 1045	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
8		3simpa 1021	. . . . . . 7 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) )
9		elnnne0 9512	. . . . . . 7 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
10	8, 9	sylibr 134	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> N e. NN )
11		nnm1nn0 9539	. . . . . 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 8596	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) = 0 <-> N = 1 ) )
14	13	biimpd 144	. . . . . . . 8 |- ( N e. NN0 -> ( ( N - 1 ) = 0 -> N = 1 ) )
15	14	necon3d 2458	. . . . . . 7 |- ( N e. NN0 -> ( N =/= 1 -> ( N - 1 ) =/= 0 ) )
16	15	imp 124	. . . . . 6 |- ( ( N e. NN0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
17	16	3adant2 1043	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
18		elnnne0 9512	. . . . 5 |- ( ( N - 1 ) e. NN <-> ( ( N - 1 ) e. NN0 /\ ( N - 1 ) =/= 0 ) )
19	12, 17, 18	sylanbrc 417	. . . 4 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN )
20		nnm1nn0 9539	. . . 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 2311	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 )
23		2nn0 9515	. . . . 5 |- 2 e. NN0
24	23	jctl 314	. . . 4 |- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) )
25	24	3ad2ant1 1045	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) )
26		nn0sub 9646	. . 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 167	1 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4111  (class class class)co 6052   0cc0 8129   1c1 8130   + caddc 8132   <_ cle 8311   - cmin 8446   NNcn 9239   2c2 9290   NN0cn0 9498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-n0 9499  df-z 9580

This theorem is referenced by:  nn0n0n1ge2b  9660  umgrclwwlkge2  16414