elznn0nn - Intuitionistic Logic Explorer

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

Theorem elznn0nn 9331

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 9319	. 2 $/\$ $\/$ $\/$
2		andi 819	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 981	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9249	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9242	. . . . . . 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 2164

cr 7871

cc0 7872

cneg 8191

cn 8982

cn0 9240

cz 9317

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-sep 4147 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-i2m1 7977 ax-rnegex 7981

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-iota 5215 df-fv 5262 df-ov 5921 df-neg 8193 df-inn 8983 df-n0 9241 df-z 9318

This theorem is referenced by: peano2z 9353 zindd 9435 expcl2lemap 10622 mulexpzap 10650 expaddzap 10654 expmulzap 10656 absexpzap 11224 pcid 12462 mulgsubcl 13206 mulgneg 13210 ghmmulg 13326