nn0le2is012 - Mathbox for Alexander van der Vekens

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Theorem nn0le2is012 30906

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		nn0re 10689	. . . 4
2		2re 10492	. . . . 5
3	2	a1i 11	. . . 4
4	1, 3	leloed 9618	. . 3 $\/$
5		nn0z 10770	. . . . . . . . 9
6		2z 10779	. . . . . . . . 9
7		zltlem1 10798	. . . . . . . . 9 $/\$
8	5, 6, 7	sylancl 662	. . . . . . . 8
9		2m1e1 10537	. . . . . . . . . 10
10	9	a1i 11	. . . . . . . . 9
11	10	breq2d 4402	. . . . . . . 8
12	8, 11	bitrd 253	. . . . . . 7
13		1red 9502	. . . . . . . . 9
14	1, 13	leloed 9618	. . . . . . . 8 $\/$
15		nn0lt10b 10807	. . . . . . . . . . . 12
16		3mix1 1157	. . . . . . . . . . . 12 $\/$ $\/$
17	15, 16	syl6bi 228	. . . . . . . . . . 11 $\/$ $\/$
18	17	com12 31	. . . . . . . . . 10 $\/$ $\/$
19		3mix2 1158	. . . . . . . . . . 11 $\/$ $\/$
20	19	a1d 25	. . . . . . . . . 10 $\/$ $\/$
21	18, 20	jaoi 379	. . . . . . . . 9 $\/$ $\/$ $\/$
22	21	com12 31	. . . . . . . 8 $\/$ $\/$ $\/$
23	14, 22	sylbid 215	. . . . . . 7 $\/$ $\/$
24	12, 23	sylbid 215	. . . . . 6 $\/$ $\/$
25	24	com12 31	. . . . 5 $\/$ $\/$
26		3mix3 1159	. . . . . 6 $\/$ $\/$
27	26	a1d 25	. . . . 5 $\/$ $\/$
28	25, 27	jaoi 379	. . . 4 $\/$ $\/$ $\/$
29	28	com12 31	. . 3 $\/$ $\/$ $\/$
30	4, 29	sylbid 215	. 2 $\/$ $\/$
31	30	imp 429	1 $/\$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 184 $\/$ wo 368 $/\$ wa 369 $\/$ w3o 964

wceq 1370

wcel 1758 class class class wbr 4390 (class class class)co 6190

cr 9382

cc0 9383

c1 9384

clt 9519

cle 9520

cmin 9696

c2 10472

cn0 10680

cz 10747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 ax-resscn 9440 ax-1cn 9441 ax-icn 9442 ax-addcl 9443 ax-addrcl 9444 ax-mulcl 9445 ax-mulrcl 9446 ax-mulcom 9447 ax-addass 9448 ax-mulass 9449 ax-distr 9450 ax-i2m1 9451 ax-1ne0 9452 ax-1rid 9453 ax-rnegex 9454 ax-rrecex 9455 ax-cnre 9456 ax-pre-lttri 9457 ax-pre-lttrn 9458 ax-pre-ltadd 9459 ax-pre-mulgt0 9460

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-pss 3442 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-tp 3980 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-tr 4484 df-eprel 4730 df-id 4734 df-po 4739 df-so 4740 df-fr 4777 df-we 4779 df-ord 4820 df-on 4821 df-lim 4822 df-suc 4823 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-riota 6151 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-om 6577 df-recs 6932 df-rdg 6966 df-er 7201 df-en 7411 df-dom 7412 df-sdom 7413 df-pnf 9521 df-mnf 9522 df-xr 9523 df-ltxr 9524 df-le 9525 df-sub 9698 df-neg 9699 df-nn 10424 df-2 10481 df-n0 10681 df-z 10748

This theorem is referenced by: exple2lt6 30907