elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 8365

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 8353	. 2 $/\$ $\/$ $\/$
2		andi 764	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 920	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 444	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 8297	. . . . . 6
6	5	pm4.71ri 384	. . . . 5 $/\$
7		elnn0 8290	. . . . . . 7 $\/$
8		orcom 679	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 182	. . . . . 6 $\/$
10	9	anbi2i 444	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 182	. . . 4 $/\$ $\/$
12	11	orbi1i 712	. . 3 $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
13	2, 4, 12	3bitr4i 210	. 2 $/\$ $\/$ $\/$ $\/$ $/\$
14	1, 13	bitri 182	1 $\/$ $/\$

Colors of variables: wff set class

Syntax hints: $/\$ wa 102

wb 103 $\/$ wo 661 $\/$ w3o 918

wceq 1284

wcel 1433

cr 6980

cc0 6981

cneg 7280

cn 8039

cn0 8288

cz 8351

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-i2m1 7081 ax-rnegex 7085

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352

This theorem is referenced by: peano2z 8387 zindd 8465 expcl2lemap 9488 mulexpzap 9516 expaddzap 9520 expmulzap 9522 absexpzap 9966