elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9421

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 9409	. 2 $/\$ $\/$ $\/$
2		andi 820	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 982	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9339	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9332	. . . . . . 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 2178

cr 7959

cc0 7960

cneg 8279

cn 9071

cn0 9330

cz 9407

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-ext 2189 ax-sep 4178 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-i2m1 8065 ax-rnegex 8069

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-br 4060 df-iota 5251 df-fv 5298 df-ov 5970 df-neg 8281 df-inn 9072 df-n0 9331 df-z 9408

This theorem is referenced by: peano2z 9443 zindd 9526 expcl2lemap 10733 mulexpzap 10761 expaddzap 10765 expmulzap 10767 absexpzap 11506 bitsfzo 12381 pcid 12762 mulgsubcl 13587 mulgneg 13591 ghmmulg 13707