nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9331

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 9269	. . . 4
2		2z 9277	. . . 4
3		zleloe 9296	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 413	. . 3 $\/$
5		zltlem1 9306	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 413	. . . . . . . 8
7		2m1e1 9033	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 4014	. . . . . . . 8
10	6, 9	bitrd 188	. . . . . . 7
11		1z 9275	. . . . . . . . 9
12		zleloe 9296	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 413	. . . . . . . 8 $\/$
14		nn0lt10b 9329	. . . . . . . . . . . 12
15		3mix1 1166	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	syl6bi 163	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1167	. . . . . . . . . . 11 $\/$ $\/$
19	18	a1d 22	. . . . . . . . . 10 $\/$ $\/$
20	17, 19	jaoi 716	. . . . . . . . 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 1168	. . . . . 6 $\/$ $\/$
26	25	a1d 22	. . . . 5 $\/$ $\/$
27	24, 26	jaoi 716	. . . 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 708 $\/$ w3o 977

wceq 1353

wcel 2148 class class class wbr 4002 (class class class)co 5872

cc0 7808

c1 7809

clt 7988

cle 7989

cmin 8124

c2 8966

cn0 9172

cz 9249

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-2 8974 df-n0 9173 df-z 9250

This theorem is referenced by: xnn0le2is012 9862