Theorem nn0n0n1ge2b 9704

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 9694	. . 3 |- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
2	1	3expib 1237	. 2 |- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) -> 2 <_ N ) )
3		nn0z 9643	. . . . . 6 |- ( N e. NN0 -> N e. ZZ )
4		0z 9634	. . . . . 6 |- 0 e. ZZ
5		zdceq 9699	. . . . . 6 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
6	3, 4, 5	sylancl 417	. . . . 5 |- ( N e. NN0 -> DECID N = 0 )
7	6	dcned 2426	. . . 4 |- ( N e. NN0 -> DECID N =/= 0 )
8		1z 9649	. . . . . 6 |- 1 e. ZZ
9		zdceq 9699	. . . . . 6 |- ( ( N e. ZZ /\ 1 e. ZZ ) -> DECID N = 1 )
10	3, 8, 9	sylancl 417	. . . . 5 |- ( N e. NN0 -> DECID N = 1 )
11	10	dcned 2426	. . . 4 |- ( N e. NN0 -> DECID N =/= 1 )
12	7, 11	dcand 945	. . 3 |- ( N e. NN0 -> DECID ( N =/= 0 /\ N =/= 1 ) )
13		ianordc 911	. . . . . 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 2425	. . . . . . 7 |- (DECID N = 0 -> ( -. N =/= 0 <-> N = 0 ) )
16	6, 15	syl 14	. . . . . 6 |- ( N e. NN0 -> ( -. N =/= 0 <-> N = 0 ) )
17		nnedc 2425	. . . . . . 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 805	. . . . 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 9374	. . . . . . . . . 10 |- 0 < 2
22		breq1 4128	. . . . . . . . . 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 9453	. . . . . . . . . 10 |- 1 < 2
26		breq1 4128	. . . . . . . . . 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 728	. . . . . . 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 9651	. . . . . . . 8 |- 2 e. ZZ
32		zltnle 9669	. . . . . . . 8 |- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> -. 2 <_ N ) )
33	3, 31, 32	sylancl 417	. . . . . . 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 865	. . 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125   0cc0 8169   1c1 8170   < clt 8350   <_ cle 8351   2c2 9334   NN0cn0 9542   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624

This theorem is referenced by: (None)