elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 8825

Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)

**Assertion**
Ref	Expression
elznn0nn	$\/$ $/\$

**Proof of Theorem elznn0nn**
Step	Hyp	Ref	Expression
1		elz 8813	. 2 $/\$ $\/$ $\/$
2		andi 768	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 926	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 446	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 8743	. . . . . 6
6	5	pm4.71ri 385	. . . . 5 $/\$
7		elnn0 8736	. . . . . . 7 $\/$
8		orcom 683	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$
10	9	anbi2i 446	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 183	. . . 4 $/\$ $\/$
12	11	orbi1i 716	. . 3 $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
13	2, 4, 12	3bitr4i 211	. 2 $/\$ $\/$ $\/$ $\/$ $/\$
14	1, 13	bitri 183	1 $\/$ $/\$

Colors of variables: wff set class

Syntax hints: $/\$ wa 103

wb 104 $\/$ wo 665 $\/$ w3o 924

wceq 1290

wcel 1439

cr 7410

cc0 7411

cneg 7715

cn 8483

cn0 8734

cz 8811

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-i2m1 7511 ax-rnegex 7515

This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812

This theorem is referenced by: peano2z 8847 zindd 8925 expcl2lemap 10028 mulexpzap 10056 expaddzap 10060 expmulzap 10062 absexpzap 10574