elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9267

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 9255	. 2 $/\$ $\/$ $\/$
2		andi 818	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 979	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9185	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9178	. . . . . . 7 $\/$
8		orcom 728	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$
10	9	anbi2i 457	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 184	. . . 4 $/\$ $\/$
12	11	orbi1i 763	. . 3 $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
13	2, 4, 12	3bitr4i 212	. 2 $/\$ $\/$ $\/$ $\/$ $/\$
14	1, 13	bitri 184	1 $\/$ $/\$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 708 $\/$ w3o 977

wceq 1353

wcel 2148

cr 7810

cc0 7811

cneg 8129

cn 8919

cn0 9176

cz 9253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4122 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-i2m1 7916 ax-rnegex 7920

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-iota 5179 df-fv 5225 df-ov 5878 df-neg 8131 df-inn 8920 df-n0 9177 df-z 9254

This theorem is referenced by: peano2z 9289 zindd 9371 expcl2lemap 10532 mulexpzap 10560 expaddzap 10564 expmulzap 10566 absexpzap 11089 pcid 12323 mulgsubcl 12997 mulgneg 13001