elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9608

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 9596	. 2 $/\$ $\/$ $\/$
2		andi 826	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9522	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9515	. . . . . . 7 $\/$
8		orcom 736	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$
10	9	anbi2i 457	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 184	. . . 4 $/\$ $\/$
12	11	orbi1i 771	. . 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 716 $\/$ w3o 1004

wceq 1398

wcel 2205

cr 8142

cc0 8143

cneg 8461

cn 9254

cn0 9513

cz 9594

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-ext 2216 ax-sep 4233 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-i2m1 8248 ax-rnegex 8252

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-un 3218 df-in 3220 df-ss 3227 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-br 4115 df-iota 5317 df-fv 5365 df-ov 6061 df-neg 8463 df-inn 9255 df-n0 9514 df-z 9595

This theorem is referenced by: peano2z 9630 zindd 9714 expcl2lemap 10937 mulexpzap 10965 expaddzap 10969 expmulzap 10971 absexpzap 11790 bitsfzo 12666 pcid 13047 mulgsubcl 13889 mulgneg 13893 ghmmulg 14009