Theorem nn0n0n1ge2b 9291

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 9282	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
2	1	3expib 1201	. 2 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) )
3		nn0z 9232	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		0z 9223	. . . . . 6 |- 0 e. ZZ
5		zdceq 9287	. . . . . 6 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
6	3, 4, 5	sylancl 411	. . . . 5 |- ( N e. NN0 -> DECID N = 0 )
7	6	dcned 2346	. . . 4 |- ( N e. NN0 -> DECID N =/= 0 )
8		1z 9238	. . . . . 6 |- 1 e. ZZ
9		zdceq 9287	. . . . . 6 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
10	3, 8, 9	sylancl 411	. . . . 5 |- ( N e. NN0 -> DECID N = 1 )
11	10	dcned 2346	. . . 4 |- ( N e. NN0 -> DECID N =/= 1 )
12		dcan2 929	. . . 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 894	. . . . . 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 2345	. . . . . . 7 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
17	6, 16	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 0 <-> N = 0 ) )
18		nnedc 2345	. . . . . . 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 788	. . . . 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 8969	. . . . . . . . . 10 |- 0 < 2
23		breq1 3992	. . . . . . . . . 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 9047	. . . . . . . . . 10 |- 1 < 2
27		breq1 3992	. . . . . . . . . 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 711	. . . . . . 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 9240	. . . . . . . 8 |- 2 e. ZZ
33		zltnle 9258	. . . . . . . 8 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> -. 2 <_ N ) )
34	3, 32, 33	sylancl 411	. . . . . . 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 848	. . 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 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   0cc0 7774   1c1 7775   < clt 7954   <_ cle 7955   2c2 8929   NN0cn0 9135   ZZcz 9212

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213

This theorem is referenced by: (None)