elznn0nn - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > elznn0nn		Unicode version

Theorem elznn0nn 9201

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 9189	. 2 $/\$ $\/$ $\/$
2		andi 808	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 969	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 453	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9119	. . . . . 6
6	5	pm4.71ri 390	. . . . 5 $/\$
7		elnn0 9112	. . . . . . 7 $\/$
8		orcom 718	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$
10	9	anbi2i 453	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 183	. . . 4 $/\$ $\/$
12	11	orbi1i 753	. . 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 698 $\/$ w3o 967

wceq 1343

wcel 2136

cr 7748

cc0 7749

cneg 8066

cn 8853

cn0 9110

cz 9187

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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 ax-sep 4099 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-i2m1 7854 ax-rnegex 7858

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ral 2448 df-rex 2449 df-rab 2452 df-v 2727 df-un 3119 df-in 3121 df-ss 3128 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-br 3982 df-iota 5152 df-fv 5195 df-ov 5844 df-neg 8068 df-inn 8854 df-n0 9111 df-z 9188

This theorem is referenced by: peano2z 9223 zindd 9305 expcl2lemap 10463 mulexpzap 10491 expaddzap 10495 expmulzap 10497 absexpzap 11018 pcid 12251