nnnn0addcl - Intuitionistic Logic Explorer

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Theorem nnnn0addcl 8318

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 8290	. 2 $\/$
2		nnaddcl 8059	. . 3 $/\$
3		oveq2 5540	. . . . 5
4		nncn 8047	. . . . . 6
5	4	addid1d 7257	. . . . 5
6	3, 5	sylan9eqr 2135	. . . 4 $/\$
7		simpl 107	. . . 4 $/\$
8	6, 7	eqeltrd 2155	. . 3 $/\$
9	2, 8	jaodan 743	. 2 $/\$ $\/$
10	1, 9	sylan2b 281	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102 $\/$ wo 661

wceq 1284

wcel 1433 (class class class)co 5532

cc0 6981

caddc 6984

cn 8039

cn0 8288

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addass 7078 ax-i2m1 7081 ax-0id 7084

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 df-n0 8289

This theorem is referenced by: nn0nnaddcl 8319 elz2 8419

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