Theorem nn0n0n1ge2 9697

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 9555	. . . . . 6 |- ( N e. NN0 -> N e. CC )
2		1cnd 8335	. . . . . 6 |- ( N e. NN0 -> 1 e. CC )
3	1, 2, 2	subsub4d 8661	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
4		1p1e2 9403	. . . . . 6 |- ( 1 + 1 ) = 2
5	4	oveq2i 6089	. . . . 5 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
6	3, 5	eqtr2di 2288	. . . 4 |- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
7	6	3ad2ant1 1049	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
8		3simpa 1025	. . . . . . 7 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) )
9		elnnne0 9559	. . . . . . 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 9586	. . . . . 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 8640	. . . . . . . . 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 2464	. . . . . . 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 1047	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
18		elnnne0 9559	. . . . 5 |- ( ( N - 1 ) e. NN <-> ( ( N - 1 ) e. NN0 /\ ( N - 1 ) =/= 0 ) )
19	12, 17, 18	sylanbrc 421	. . . 4 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) e. NN )
20		nnm1nn0 9586	. . . 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 2315	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 )
23		2nn0 9562	. . . . 5 |- 2 e. NN0
24	23	jctl 314	. . . 4 |- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) )
25	24	3ad2ant1 1049	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) )
26		nn0sub 9693	. . 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4128  (class class class)co 6078   0cc0 8172   1c1 8173   + caddc 8175   <_ cle 8354   - cmin 8490   NNcn 9286   2c2 9337   NN0cn0 9545

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-0id 8280  ax-rnegex 8281  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-ltadd 8288

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-inn 9287  df-2 9345  df-n0 9546  df-z 9627

This theorem is referenced by:  nn0n0n1ge2b  9707  umgrclwwlkge2  16560