elznn0nn - Intuitionistic Logic Explorer

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Theorem elznn0nn 9669

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 9657	. 2 $/\$ $\/$ $\/$
2		andi 830	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$ $/\$
3		df-3or 1010	. . . 4 $\/$ $\/$ $\/$ $\/$
4	3	anbi2i 461	. . 3 $/\$ $\/$ $\/$ $/\$ $\/$ $\/$
5		nn0re 9583	. . . . . 6
6	5	pm4.71ri 396	. . . . 5 $/\$
7		elnn0 9576	. . . . . . 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

This proof depends on syntax axioms: $/\$ wa 104

wb 105 $\/$ wo 720 $\/$ w3o 1008

wceq 1402

wcel 2209

cr 8184

cc0 8185

cneg 8505

cn 9313

cn0 9574

cz 9655

This proof depends on 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 4251 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-i2m1 8290 ax-rnegex 8294

This proof 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 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4132 df-iota 5340 df-fv 5388 df-ov 6091 df-neg 8507 df-inn 9314 df-n0 9575 df-z 9656

This theorem is used by: peano2z 9691 zindd 9775 expcl2lemap 11009 mulexpzap 11037 expaddzap 11041 expmulzap 11043 absexpzap 11869 bitsfzo 12747 pcid 13132 mulgsubcl 13998 mulgneg 14002 ghmmulg 14118