Theorem nn0n0n1ge2b 9399

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 9390	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
2	1	3expib 1208	. 2 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) )
3		nn0z 9340	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		0z 9331	. . . . . 6 |- 0 e. ZZ
5		zdceq 9395	. . . . . 6 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
6	3, 4, 5	sylancl 413	. . . . 5 |- ( N e. NN0 -> DECID N = 0 )
7	6	dcned 2370	. . . 4 |- ( N e. NN0 -> DECID N =/= 0 )
8		1z 9346	. . . . . 6 |- 1 e. ZZ
9		zdceq 9395	. . . . . 6 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
10	3, 8, 9	sylancl 413	. . . . 5 |- ( N e. NN0 -> DECID N = 1 )
11	10	dcned 2370	. . . 4 |- ( N e. NN0 -> DECID N =/= 1 )
12	7, 11	dcand 934	. . 3 |- ( N e. NN0 -> DECID ( N =/= 0 /\ N =/= 1 ) )
13		ianordc 900	. . . . . 6 |- (DECID N =/= 0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( -. N =/= 0 \/ -. N =/= 1 ) ) )
14	7, 13	syl 14	. . . . 5 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( -. N =/= 0 \/ -. N =/= 1 ) ) )
15		nnedc 2369	. . . . . . 7 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
16	6, 15	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 0 <-> N = 0 ) )
17		nnedc 2369	. . . . . . 7 |- (DECID N = 1 -> ( -. N =/= 1 <-> N = 1 ) )
18	10, 17	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 1 <-> N = 1 ) )
19	16, 18	orbi12d 794	. . . . 5 |- ( N e. NN0 -> ( ( -. N =/= 0 \/ -. N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
20	14, 19	bitrd 188	. . . 4 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) <-> ( N = 0 \/ N = 1 ) ) )
21		2pos 9075	. . . . . . . . . 10 |- 0 < 2
22		breq1 4033	. . . . . . . . . 10 |- ( N = 0 -> ( N < 2 <-> 0 < 2 ) )
23	21, 22	mpbiri 168	. . . . . . . . 9 |- ( N = 0 -> N < 2 )
24	23	a1d 22	. . . . . . . 8 |- ( N = 0 -> ( N e. NN0 -> N < 2 ) )
25		1lt2 9154	. . . . . . . . . 10 |- 1 < 2
26		breq1 4033	. . . . . . . . . 10 |- ( N = 1 -> ( N < 2 <-> 1 < 2 ) )
27	25, 26	mpbiri 168	. . . . . . . . 9 |- ( N = 1 -> N < 2 )
28	27	a1d 22	. . . . . . . 8 |- ( N = 1 -> ( N e. NN0 -> N < 2 ) )
29	24, 28	jaoi 717	. . . . . . 7 |- ( ( N = 0 \/ N = 1 ) -> ( N e. NN0 -> N < 2 ) )
30	29	impcom 125	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> N < 2 )
31		2z 9348	. . . . . . . 8 |- 2 e. ZZ
32		zltnle 9366	. . . . . . . 8 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> -. 2 <_ N ) )
33	3, 31, 32	sylancl 413	. . . . . . 7 |- ( N e. NN0 -> ( N < 2 <-> -. 2 <_ N ) )
34	33	adantr 276	. . . . . 6 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> ( N < 2 <-> -. 2 <_ N ) )
35	30, 34	mpbid 147	. . . . 5 |- ( ( N e. NN0 /\ ( N = 0 \/ N = 1 ) ) -> -. 2 <_ N )
36	35	ex 115	. . . 4 |- ( N e. NN0 -> ( ( N = 0 \/ N = 1 ) -> -. 2 <_ N ) )
37	20, 36	sylbid 150	. . 3 |- ( N e. NN0 -> ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) )
38		condc 854	. . 3 |- (DECID ( N =/= 0 /\ N =/= 1 ) -> ( ( -. ( N =/= 0 /\ N =/= 1 ) -> -. 2 <_ N ) -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) ) )
39	12, 37, 38	sylc 62	. 2 |- ( N e. NN0 -> ( 2 <_ N -> ( N =/= 0 /\ N =/= 1 ) ) )
40	2, 39	impbid 129	1 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4030   0cc0 7874   1c1 7875   < clt 8056   <_ cle 8057   2c2 9035   NN0cn0 9243   ZZcz 9320

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321

This theorem is referenced by: (None)