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Mirrors > Home > ILE Home > Th. List > nnnn0addcl | Unicode version |
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.) |
Ref | Expression |
---|---|
nnnn0addcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8979 | . 2 | |
2 | nnaddcl 8740 | . . 3 | |
3 | oveq2 5782 | . . . . 5 | |
4 | nncn 8728 | . . . . . 6 | |
5 | 4 | addid1d 7911 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2194 | . . . 4 |
7 | simpl 108 | . . . 4 | |
8 | 6, 7 | eqeltrd 2216 | . . 3 |
9 | 2, 8 | jaodan 786 | . 2 |
10 | 1, 9 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 (class class class)co 5774 cc0 7620 caddc 7623 cn 8720 cn0 8977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 df-n0 8978 |
This theorem is referenced by: nn0nnaddcl 9008 elz2 9122 bcxmas 11258 |
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