nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9666

Description: A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)

**Assertion**
Ref	Expression
nn0le2is012	$/\$ $\/$ $\/$

**Proof of Theorem nn0le2is012**
Step	Hyp	Ref	Expression
1		nn0z 9602	. . . 4
2		2z 9610	. . . 4
3		zleloe 9629	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 413	. . 3 $\/$
5		zltlem1 9640	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 413	. . . . . . . 8
7		2m1e1 9360	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 4123	. . . . . . . 8
10	6, 9	bitrd 188	. . . . . . 7
11		1z 9608	. . . . . . . . 9
12		zleloe 9629	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 413	. . . . . . . 8 $\/$
14		nn0lt10b 9664	. . . . . . . . . . . 12
15		3mix1 1193	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	biimtrdi 163	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1194	. . . . . . . . . . 11 $\/$ $\/$
19	18	a1d 22	. . . . . . . . . 10 $\/$ $\/$
20	17, 19	jaoi 724	. . . . . . . . 9 $\/$ $\/$ $\/$
21	20	com12 30	. . . . . . . 8 $\/$ $\/$ $\/$
22	13, 21	sylbid 150	. . . . . . 7 $\/$ $\/$
23	10, 22	sylbid 150	. . . . . 6 $\/$ $\/$
24	23	com12 30	. . . . 5 $\/$ $\/$
25		3mix3 1195	. . . . . 6 $\/$ $\/$
26	25	a1d 22	. . . . 5 $\/$ $\/$
27	24, 26	jaoi 724	. . . 4 $\/$ $\/$ $\/$
28	27	com12 30	. . 3 $\/$ $\/$ $\/$
29	4, 28	sylbid 150	. 2 $\/$ $\/$
30	29	imp 124	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $\/$ w3o 1004

wceq 1398

wcel 2205 class class class wbr 4111 (class class class)co 6052

cc0 8132

c1 8133

clt 8313

cle 8314

cmin 8449

c2 9293

cn0 9501

cz 9582

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4230 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-addcom 8232 ax-addass 8234 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-0id 8240 ax-rnegex 8241 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3045 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-br 4112 df-opab 4174 df-id 4416 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-iota 5314 df-fun 5356 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-inn 9243 df-2 9301 df-n0 9502 df-z 9583

This theorem is referenced by: xnn0le2is012 10205