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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9177 |
. 2
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2 | nnaddcl 8938 |
. . 3
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3 | oveq2 5882 |
. . . . 5
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4 | nncn 8926 |
. . . . . 6
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5 | 4 | addid1d 8105 |
. . . . 5
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6 | 3, 5 | sylan9eqr 2232 |
. . . 4
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7 | simpl 109 |
. . . 4
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8 | 6, 7 | eqeltrd 2254 |
. . 3
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9 | 2, 8 | jaodan 797 |
. 2
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10 | 1, 9 | sylan2b 287 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4121 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addass 7912 ax-i2m1 7915 ax-0id 7918 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-inn 8919 df-n0 9176 |
This theorem is referenced by: nn0nnaddcl 9206 elz2 9323 bcxmas 11496 |
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