elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9357

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 9345	. 2 $/\$ $\/$ $\/$
2		andi 819	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9275	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9268	. . . . . . 7 $\/$
8		orcom 729	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$
10	9	anbi2i 457	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 184	. . . 4 $/\$ $\/$
12	11	orbi1i 764	. . 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 709 $\/$ w3o 979

wceq 1364

wcel 2167

cr 7895

cc0 7896

cneg 8215

cn 9007

cn0 9266

cz 9343

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-ext 2178 ax-sep 4152 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-i2m1 8001 ax-rnegex 8005

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-br 4035 df-iota 5220 df-fv 5267 df-ov 5928 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344

This theorem is referenced by: peano2z 9379 zindd 9461 expcl2lemap 10660 mulexpzap 10688 expaddzap 10692 expmulzap 10694 absexpzap 11262 bitsfzo 12137 pcid 12518 mulgsubcl 13342 mulgneg 13346 ghmmulg 13462