nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9683

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 9619	. . . 4
2		2z 9627	. . . 4
3		zleloe 9646	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 413	. . 3 $\/$
5		zltlem1 9657	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 413	. . . . . . . 8
7		2m1e1 9377	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 4127	. . . . . . . 8
10	6, 9	bitrd 188	. . . . . . 7
11		1z 9625	. . . . . . . . 9
12		zleloe 9646	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 413	. . . . . . . 8 $\/$
14		nn0lt10b 9681	. . . . . . . . . . . 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 4115 (class class class)co 6060

cc0 8145

c1 8146

clt 8326

cle 8327

cmin 8463

c2 9310

cn0 9518

cz 9599

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-14 2208 ax-ext 2216 ax-sep 4234 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4666 ax-cnex 8236 ax-resscn 8237 ax-1cn 8238 ax-1re 8239 ax-icn 8240 ax-addcl 8241 ax-addrcl 8242 ax-mulcl 8243 ax-addcom 8245 ax-addass 8247 ax-distr 8249 ax-i2m1 8250 ax-0lt1 8251 ax-0id 8253 ax-rnegex 8254 ax-cnre 8256 ax-pre-ltirr 8257 ax-pre-ltwlin 8258 ax-pre-lttrn 8259 ax-pre-apti 8260 ax-pre-ltadd 8261

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 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-int 3956 df-br 4116 df-opab 4178 df-id 4420 df-xp 4762 df-rel 4763 df-cnv 4764 df-co 4765 df-dm 4766 df-iota 5319 df-fun 5361 df-fv 5367 df-riota 6013 df-ov 6063 df-oprab 6064 df-mpo 6065 df-pnf 8328 df-mnf 8329 df-xr 8330 df-ltxr 8331 df-le 8332 df-sub 8465 df-neg 8466 df-inn 9260 df-2 9318 df-n0 9519 df-z 9600

This theorem is referenced by: xnn0le2is012 10223