Theorem nn0n0n1ge2b 8544

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 8535	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
2	1	3expib 1142	. 2 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) )
3		nn0z 8488	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		0z 8479	. . . . . 6 |- 0 e. ZZ
5		zdceq 8540	. . . . . 6 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
6	3, 4, 5	sylancl 404	. . . . 5 |- ( N e. NN0 -> DECID N = 0 )
7	6	dcned 2255	. . . 4 |- ( N e. NN0 -> DECID N =/= 0 )
8		1z 8494	. . . . . 6 |- 1 e. ZZ
9		zdceq 8540	. . . . . 6 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
10	3, 8, 9	sylancl 404	. . . . 5 |- ( N e. NN0 -> DECID N = 1 )
11	10	dcned 2255	. . . 4 |- ( N e. NN0 -> DECID N =/= 1 )
12		dcan 876	. . . 4 |- (DECID N =/= 0 -> (DECID N =/= 1 -> DECID ( N =/= 0 /\ N =/= 1 ) ) )
13	7, 11, 12	sylc 61	. . 3 |- ( N e. NN0 -> DECID ( N =/= 0 /\ N =/= 1 ) )
14		ianordc 833	. . . . . 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 2254	. . . . . . 7 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
17	6, 16	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 0 <-> N = 0 ) )
18		nnedc 2254	. . . . . . 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 740	. . . . 5 |- ( N e. NN0 -> ( ( -. N =/= 0 \/ -. N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
21	15, 20	bitrd 186	. . . 4 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
22		2pos 8233	. . . . . . . . . 10 |- 0 < 2
23		breq1 3809	. . . . . . . . . 10 |- ( N = 0 -> ( N < 2 <-> 0 < 2 ) )
24	22, 23	mpbiri 166	. . . . . . . . 9 |- ( N = 0 -> N < 2 )
25	24	a1d 22	. . . . . . . 8 |- ( N = 0 -> ( N e. NN0 -> N < 2 ) )
26		1lt2 8304	. . . . . . . . . 10 |- 1 < 2
27		breq1 3809	. . . . . . . . . 10 |- ( N = 1 -> ( N < 2 <-> 1 < 2 ) )
28	26, 27	mpbiri 166	. . . . . . . . 9 |- ( N = 1 -> N < 2 )
29	28	a1d 22	. . . . . . . 8 |- ( N = 1 -> ( N e. NN0 -> N < 2 ) )
30	25, 29	jaoi 669	. . . . . . 7 |- ( ( N = 0 \/ N = 1 ) -> ( N e. NN0 -> N < 2 ) )
31	30	impcom 123	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> N < 2 )
32		2z 8496	. . . . . . . 8 |- 2 e. ZZ
33		zltnle 8514	. . . . . . . 8 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> -. 2 <_ N ) )
34	3, 32, 33	sylancl 404	. . . . . . 7 |- ( N e. NN0 -> ( N < 2 <-> -. 2 <_ N ) )
35	34	adantr 270	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> ( N < 2 <-> -. 2 <_ N ) )
36	31, 35	mpbid 145	. . . . 5 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> -. 2 <_ N )
37	36	ex 113	. . . 4 |- ( N e. NN0 -> ( ( N = 0 \/ N = 1 ) -> -. 2 <_ N ) )
38	21, 37	sylbid 148	. . 3 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) )
39		condc 783	. . 3 |- (DECID ( N =/= 0 /\ N =/= 1 ) -> ( ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) ) )
40	13, 38, 39	sylc 61	. 2 |- ( N e. NN0 -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) )
41	2, 40	impbid 127	1 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662  DECID wdc 776   = wceq 1285   e. wcel 1434   =/= wne 2249   class class class wbr 3806   0cc0 7079   1c1 7080   < clt 7251   <_ cle 7252   2c2 8192   NN0cn0 8391   ZZcz 8468

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-2 8201  df-n0 8392  df-z 8469

This theorem is referenced by: (None)