Theorem nn0n0n1ge2b 9270

Description: A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)

Assertion

Ref	Expression
nn0n0n1ge2b	|- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )

Proof of Theorem nn0n0n1ge2b

Step	Hyp	Ref	Expression
1		nn0n0n1ge2 9261	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
2	1	3expib 1196	. 2 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) )
3		nn0z 9211	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		0z 9202	. . . . . 6 |- 0 e. ZZ
5		zdceq 9266	. . . . . 6 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
6	3, 4, 5	sylancl 410	. . . . 5 |- ( N e. NN0 -> DECID N = 0 )
7	6	dcned 2342	. . . 4 |- ( N e. NN0 -> DECID N =/= 0 )
8		1z 9217	. . . . . 6 |- 1 e. ZZ
9		zdceq 9266	. . . . . 6 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
10	3, 8, 9	sylancl 410	. . . . 5 |- ( N e. NN0 -> DECID N = 1 )
11	10	dcned 2342	. . . 4 |- ( N e. NN0 -> DECID N =/= 1 )
12		dcan2 924	. . . 4 |- (DECID N =/= 0 -> (DECID N =/= 1 -> DECID ( N =/= 0 /\ N =/= 1 ) ) )
13	7, 11, 12	sylc 62	. . 3 |- ( N e. NN0 -> DECID ( N =/= 0 /\ N =/= 1 ) )
14		ianordc 889	. . . . . 6 |- (DECID N =/= 0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( -. N =/= 0 \/ -. N =/= 1 ) ) )
15	7, 14	syl 14	. . . . 5 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( -. N =/= 0 \/ -. N =/= 1 ) ) )
16		nnedc 2341	. . . . . . 7 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
17	6, 16	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 0 <-> N = 0 ) )
18		nnedc 2341	. . . . . . 7 |- (DECID N = 1 -> ( -. N =/= 1 <-> N = 1 ) )
19	10, 18	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 1 <-> N = 1 ) )
20	17, 19	orbi12d 783	. . . . 5 |- ( N e. NN0 -> ( ( -. N =/= 0 \/ -. N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
21	15, 20	bitrd 187	. . . 4 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
22		2pos 8948	. . . . . . . . . 10 |- 0 < 2
23		breq1 3985	. . . . . . . . . 10 |- ( N = 0 -> ( N < 2 <-> 0 < 2 ) )
24	22, 23	mpbiri 167	. . . . . . . . 9 |- ( N = 0 -> N < 2 )
25	24	a1d 22	. . . . . . . 8 |- ( N = 0 -> ( N e. NN0 -> N < 2 ) )
26		1lt2 9026	. . . . . . . . . 10 |- 1 < 2
27		breq1 3985	. . . . . . . . . 10 |- ( N = 1 -> ( N < 2 <-> 1 < 2 ) )
28	26, 27	mpbiri 167	. . . . . . . . 9 |- ( N = 1 -> N < 2 )
29	28	a1d 22	. . . . . . . 8 |- ( N = 1 -> ( N e. NN0 -> N < 2 ) )
30	25, 29	jaoi 706	. . . . . . 7 |- ( ( N = 0 \/ N = 1 ) -> ( N e. NN0 -> N < 2 ) )
31	30	impcom 124	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> N < 2 )
32		2z 9219	. . . . . . . 8 |- 2 e. ZZ
33		zltnle 9237	. . . . . . . 8 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> -. 2 <_ N ) )
34	3, 32, 33	sylancl 410	. . . . . . 7 |- ( N e. NN0 -> ( N < 2 <-> -. 2 <_ N ) )
35	34	adantr 274	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> ( N < 2 <-> -. 2 <_ N ) )
36	31, 35	mpbid 146	. . . . 5 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> -. 2 <_ N )
37	36	ex 114	. . . 4 |- ( N e. NN0 -> ( ( N = 0 \/ N = 1 ) -> -. 2 <_ N ) )
38	21, 37	sylbid 149	. . 3 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) )
39		condc 843	. . 3 |- (DECID ( N =/= 0 /\ N =/= 1 ) -> ( ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) ) )
40	13, 38, 39	sylc 62	. 2 |- ( N e. NN0 -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) )
41	2, 40	impbid 128	1 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   0cc0 7753   1c1 7754   < clt 7933   <_ cle 7934   2c2 8908   NN0cn0 9114   ZZcz 9191

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-stab 821  df-dc 825  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: (None)