nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9273

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 9211	. . . 4
2		2z 9219	. . . 4
3		zleloe 9238	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 410	. . 3 $\/$
5		zltlem1 9248	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 410	. . . . . . . 8
7		2m1e1 8975	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 3994	. . . . . . . 8
10	6, 9	bitrd 187	. . . . . . 7
11		1z 9217	. . . . . . . . 9
12		zleloe 9238	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 410	. . . . . . . 8 $\/$
14		nn0lt10b 9271	. . . . . . . . . . . 12
15		3mix1 1156	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	syl6bi 162	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1157	. . . . . . . . . . 11 $\/$ $\/$
19	18	a1d 22	. . . . . . . . . 10 $\/$ $\/$
20	17, 19	jaoi 706	. . . . . . . . 9 $\/$ $\/$ $\/$
21	20	com12 30	. . . . . . . 8 $\/$ $\/$ $\/$
22	13, 21	sylbid 149	. . . . . . 7 $\/$ $\/$
23	10, 22	sylbid 149	. . . . . 6 $\/$ $\/$
24	23	com12 30	. . . . 5 $\/$ $\/$
25		3mix3 1158	. . . . . 6 $\/$ $\/$
26	25	a1d 22	. . . . 5 $\/$ $\/$
27	24, 26	jaoi 706	. . . 4 $\/$ $\/$ $\/$
28	27	com12 30	. . 3 $\/$ $\/$ $\/$
29	4, 28	sylbid 149	. 2 $\/$ $\/$
30	29	imp 123	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $\/$ w3o 967

wceq 1343

wcel 2136 class class class wbr 3982 (class class class)co 5842

cc0 7753

c1 7754

clt 7933

cle 7934

cmin 8069

c2 8908

cn0 9114

cz 9191

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-inn 8858 df-2 8916 df-n0 9115 df-z 9192

This theorem is referenced by: xnn0le2is012 9802