elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9388

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 9376	. 2 $/\$ $\/$ $\/$
2		andi 820	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 982	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9306	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9299	. . . . . . 7 $\/$
8		orcom 730	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$
10	9	anbi2i 457	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 184	. . . 4 $/\$ $\/$
12	11	orbi1i 765	. . 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 710 $\/$ w3o 980

wceq 1373

wcel 2176

cr 7926

cc0 7927

cneg 8246

cn 9038

cn0 9297

cz 9374

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 ax-sep 4163 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-i2m1 8032 ax-rnegex 8036

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4046 df-iota 5233 df-fv 5280 df-ov 5949 df-neg 8248 df-inn 9039 df-n0 9298 df-z 9375

This theorem is referenced by: peano2z 9410 zindd 9493 expcl2lemap 10698 mulexpzap 10726 expaddzap 10730 expmulzap 10732 absexpzap 11424 bitsfzo 12299 pcid 12680 mulgsubcl 13505 mulgneg 13509 ghmmulg 13625