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Mirrors > Home > ILE Home > Th. List > nn0nnaddcl | Unicode version |
Description: A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Ref | Expression |
---|---|
nn0nnaddcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 7922 | . . . 4 | |
2 | nn0cn 8191 | . . . 4 | |
3 | addcom 7150 | . . . 4 | |
4 | 1, 2, 3 | syl2an 273 | . . 3 |
5 | nnnn0addcl 8212 | . . 3 | |
6 | 4, 5 | eqeltrrd 2115 | . 2 |
7 | 6 | ancoms 255 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 (class class class)co 5512 cc 6887 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-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 |
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: nn0p1nn 8221 nnaddm1cl 8305 numnncl 8375 |
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