nnnn0addcl - Intuitionistic Logic Explorer

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Theorem nnnn0addcl 8212

Description: A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)

**Assertion**
Ref	Expression
nnnn0addcl	$/\$

**Proof of Theorem nnnn0addcl**
Step	Hyp	Ref	Expression
1		elnn0 8183	. 2 $\/$
2		nnaddcl 7934	. . 3 $/\$
3		oveq2 5520	. . . . 5
4		nncn 7922	. . . . . 6
5	4	addid1d 7162	. . . . 5
6	3, 5	sylan9eqr 2094	. . . 4 $/\$
7		simpl 102	. . . 4 $/\$
8	6, 7	eqeltrd 2114	. . 3 $/\$
9	2, 8	jaodan 710	. 2 $/\$ $\/$
10	1, 9	sylan2b 271	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97 $\/$ wo 629

wceq 1243

wcel 1393 (class class class)co 5512

cc0 6889

caddc 6892

cn 7914

cn0 8181

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addass 6986 ax-i2m1 6989 ax-0id 6992

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-inn 7915 df-n0 8182

This theorem is referenced by: nn0nnaddcl 8213 elz2 8312

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