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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 8972 | . 2 | |
2 | nnaddcl 8733 | . . 3 | |
3 | oveq2 5775 | . . . . 5 | |
4 | nncn 8721 | . . . . . 6 | |
5 | 4 | addid1d 7904 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2192 | . . . 4 |
7 | simpl 108 | . . . 4 | |
8 | 6, 7 | eqeltrd 2214 | . . 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 5767 cc0 7613 caddc 7616 cn 8713 cn0 8970 |
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 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 df-n0 8971 |
This theorem is referenced by: nn0nnaddcl 9001 elz2 9115 bcxmas 11251 |
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