nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9157

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 9098	. . . 4
2		2z 9106	. . . 4
3		zleloe 9125	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 410	. . 3 $\/$
5		zltlem1 9135	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 410	. . . . . . . 8
7		2m1e1 8862	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 3949	. . . . . . . 8
10	6, 9	bitrd 187	. . . . . . 7
11		1z 9104	. . . . . . . . 9
12		zleloe 9125	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 410	. . . . . . . 8 $\/$
14		nn0lt10b 9155	. . . . . . . . . . . 12
15		3mix1 1151	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	syl6bi 162	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1152	. . . . . . . . . . 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 1153	. . . . . 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 962

wceq 1332

wcel 1481 class class class wbr 3937 (class class class)co 5782

cc0 7644

c1 7645

clt 7824

cle 7825

cmin 7957

c2 8795

cn0 9001

cz 9078

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079

This theorem is referenced by: xnn0le2is012 9679