nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9133

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 9074	. . . 4
2		2z 9082	. . . 4
3		zleloe 9101	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 409	. . 3 $\/$
5		zltlem1 9111	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 409	. . . . . . . 8
7		2m1e1 8838	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 3941	. . . . . . . 8
10	6, 9	bitrd 187	. . . . . . 7
11		1z 9080	. . . . . . . . 9
12		zleloe 9101	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 409	. . . . . . . 8 $\/$
14		nn0lt10b 9131	. . . . . . . . . . . 12
15		3mix1 1150	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	syl6bi 162	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1151	. . . . . . . . . . 11 $\/$ $\/$
19	18	a1d 22	. . . . . . . . . 10 $\/$ $\/$
20	17, 19	jaoi 705	. . . . . . . . 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 1152	. . . . . 6 $\/$ $\/$
26	25	a1d 22	. . . . 5 $\/$ $\/$
27	24, 26	jaoi 705	. . . 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 697 $\/$ w3o 961

wceq 1331

wcel 1480 class class class wbr 3929 (class class class)co 5774

cc0 7620

c1 7621

clt 7800

cle 7801

cmin 7933

c2 8771

cn0 8977

cz 9054

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055

This theorem is referenced by: xnn0le2is012 9649