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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 9137 | . 2 | |
2 | nnaddcl 8898 | . . 3 | |
3 | oveq2 5861 | . . . . 5 | |
4 | nncn 8886 | . . . . . 6 | |
5 | 4 | addid1d 8068 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2225 | . . . 4 |
7 | simpl 108 | . . . 4 | |
8 | 6, 7 | eqeltrd 2247 | . . 3 |
9 | 2, 8 | jaodan 792 | . 2 |
10 | 1, 9 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 (class class class)co 5853 cc0 7774 caddc 7777 cn 8878 cn0 9135 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4107 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addass 7876 ax-i2m1 7879 ax-0id 7882 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-inn 8879 df-n0 9136 |
This theorem is referenced by: nn0nnaddcl 9166 elz2 9283 bcxmas 11452 |
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