Theorem nn0n0n1ge2 9325

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 9188	. . . . . 6 |- ( N e. NN0 -> N e. CC )
2		1cnd 7975	. . . . . 6 |- ( N e. NN0 -> 1 e. CC )
3	1, 2, 2	subsub4d 8301	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
4		1p1e2 9038	. . . . . 6 |- ( 1 + 1 ) = 2
5	4	oveq2i 5888	. . . . 5 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
6	3, 5	eqtr2di 2227	. . . 4 |- ( N e. NN0 -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
7	6	3ad2ant1 1018	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) = ( ( N - 1 ) - 1 ) )
8		3simpa 994	. . . . . . 7 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N e. NN0 /\ N =/= 0 ) )
9		elnnne0 9192	. . . . . . 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 9219	. . . . . 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 8280	. . . . . . . . 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 2391	. . . . . . 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 1016	. . . . 5 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
18		elnnne0 9192	. . . . 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 9219	. . . 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 2254	. 2 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( N - 2 ) e. NN0 )
23		2nn0 9195	. . . . 5 |- 2 e. NN0
24	23	jctl 314	. . . 4 |- ( N e. NN0 -> ( 2 e. NN0 /\ N e. NN0 ) )
25	24	3ad2ant1 1018	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> ( 2 e. NN0 /\ N e. NN0 ) )
26		nn0sub 9321	. . 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 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005  (class class class)co 5877   0cc0 7813   1c1 7814   + caddc 7816   <_ cle 7995   - cmin 8130   NNcn 8921   2c2 8972   NN0cn0 9178

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256

This theorem is referenced by:  nn0n0n1ge2b  9334