elznn0nn - Intuitionistic Logic Explorer

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

Theorem elznn0nn 9637

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 9625	. 2 $/\$ $\/$ $\/$
2		andi 830	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 1010	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 461	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9551	. . . . . 6
6	5	pm4.71ri 396	. . . . 5 $/\$
7		elnn0 9544	. . . . . . 7 $\/$
8		orcom 740	. . . . . . 7 $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$
10	9	anbi2i 461	. . . . 5 $/\$ $/\$ $\/$
11	6, 10	bitri 184	. . . 4 $/\$ $\/$
12	11	orbi1i 775	. . 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 720 $\/$ w3o 1008

wceq 1402

wcel 2209

cr 8168

cc0 8169

cneg 8488

cn 9283

cn0 9542

cz 9623

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 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220 ax-sep 4244 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-i2m1 8274 ax-rnegex 8278

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-br 4126 df-iota 5332 df-fv 5380 df-ov 6078 df-neg 8490 df-inn 9284 df-n0 9543 df-z 9624

This theorem is referenced by: peano2z 9659 zindd 9743 expcl2lemap 10966 mulexpzap 10994 expaddzap 10998 expmulzap 11000 absexpzap 11824 bitsfzo 12700 pcid 13081 mulgsubcl 13916 mulgneg 13920 ghmmulg 14036