elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9068

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 9056	. 2 $/\$ $\/$ $\/$
2		andi 807	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 963	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 452	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 8986	. . . . . 6
6	5	pm4.71ri 389	. . . . 5 $/\$
7		elnn0 8979	. . . . . . 7 $\/$
8		orcom 717	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$
10	9	anbi2i 452	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 183	. . . 4 $/\$ $\/$
12	11	orbi1i 752	. . 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 697 $\/$ w3o 961

wceq 1331

wcel 1480

cr 7619

cc0 7620

cneg 7934

cn 8720

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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 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-i2m1 7725 ax-rnegex 7729

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055

This theorem is referenced by: peano2z 9090 zindd 9169 expcl2lemap 10305 mulexpzap 10333 expaddzap 10337 expmulzap 10339 absexpzap 10852