nn0le2is012 - Intuitionistic Logic Explorer

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Theorem nn0le2is012 9294

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 9232	. . . 4
2		2z 9240	. . . 4
3		zleloe 9259	. . . 4 $/\$ $\/$
4	1, 2, 3	sylancl 411	. . 3 $\/$
5		zltlem1 9269	. . . . . . . . 9 $/\$
6	1, 2, 5	sylancl 411	. . . . . . . 8
7		2m1e1 8996	. . . . . . . . . 10
8	7	a1i 9	. . . . . . . . 9
9	8	breq2d 4001	. . . . . . . 8
10	6, 9	bitrd 187	. . . . . . 7
11		1z 9238	. . . . . . . . 9
12		zleloe 9259	. . . . . . . . 9 $/\$ $\/$
13	1, 11, 12	sylancl 411	. . . . . . . 8 $\/$
14		nn0lt10b 9292	. . . . . . . . . . . 12
15		3mix1 1161	. . . . . . . . . . . 12 $\/$ $\/$
16	14, 15	syl6bi 162	. . . . . . . . . . 11 $\/$ $\/$
17	16	com12 30	. . . . . . . . . 10 $\/$ $\/$
18		3mix2 1162	. . . . . . . . . . 11 $\/$ $\/$
19	18	a1d 22	. . . . . . . . . 10 $\/$ $\/$
20	17, 19	jaoi 711	. . . . . . . . 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 1163	. . . . . 6 $\/$ $\/$
26	25	a1d 22	. . . . 5 $\/$ $\/$
27	24, 26	jaoi 711	. . . 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 703 $\/$ w3o 972

wceq 1348

wcel 2141 class class class wbr 3989 (class class class)co 5853

cc0 7774

c1 7775

clt 7954

cle 7955

cmin 8090

c2 8929

cn0 9135

cz 9212

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213

This theorem is referenced by: xnn0le2is012 9823