elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9226

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 9214	. 2 $/\$ $\/$ $\/$
2		andi 813	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 974	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 454	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9144	. . . . . 6
6	5	pm4.71ri 390	. . . . 5 $/\$
7		elnn0 9137	. . . . . . 7 $\/$
8		orcom 723	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$
10	9	anbi2i 454	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 183	. . . 4 $/\$ $\/$
12	11	orbi1i 758	. . 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 703 $\/$ w3o 972

wceq 1348

wcel 2141

cr 7773

cc0 7774

cneg 8091

cn 8878

cn0 9135

cz 9212

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-i2m1 7879 ax-rnegex 7883

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213

This theorem is referenced by: peano2z 9248 zindd 9330 expcl2lemap 10488 mulexpzap 10516 expaddzap 10520 expmulzap 10522 absexpzap 11044 pcid 12277