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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 9092 | . 2 | |
2 | nnaddcl 8853 | . . 3 | |
3 | oveq2 5832 | . . . . 5 | |
4 | nncn 8841 | . . . . . 6 | |
5 | 4 | addid1d 8024 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2212 | . . . 4 |
7 | simpl 108 | . . . 4 | |
8 | 6, 7 | eqeltrd 2234 | . . 3 |
9 | 2, 8 | jaodan 787 | . 2 |
10 | 1, 9 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 (class class class)co 5824 cc0 7732 caddc 7735 cn 8833 cn0 9090 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-sep 4082 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addass 7834 ax-i2m1 7837 ax-0id 7840 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 df-inn 8834 df-n0 9091 |
This theorem is referenced by: nn0nnaddcl 9121 elz2 9235 bcxmas 11386 |
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