elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9591

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 9579	. 2 $/\$ $\/$ $\/$
2		andi 826	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 1006	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 457	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9505	. . . . . 6
6	5	pm4.71ri 392	. . . . 5 $/\$
7		elnn0 9498	. . . . . . 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 2203

cr 8126

cc0 8127

cneg 8445

cn 9237

cn0 9496

cz 9577

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 2214 ax-sep 4228 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-i2m1 8232 ax-rnegex 8236

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 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-un 3215 df-in 3217 df-ss 3224 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-br 4110 df-iota 5312 df-fv 5360 df-ov 6053 df-neg 8447 df-inn 9238 df-n0 9497 df-z 9578

This theorem is referenced by: peano2z 9613 zindd 9696 expcl2lemap 10913 mulexpzap 10941 expaddzap 10945 expmulzap 10947 absexpzap 11765 bitsfzo 12641 pcid 13022 mulgsubcl 13853 mulgneg 13857 ghmmulg 13973