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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 9178 |
. 2
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2 | nnaddcl 8939 |
. . 3
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3 | oveq2 5883 |
. . . . 5
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4 | nncn 8927 |
. . . . . 6
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5 | 4 | addid1d 8106 |
. . . . 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 4122 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addass 7913 ax-i2m1 7916 ax-0id 7919 |
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 2740 df-un 3134 df-in 3136 df-ss 3143 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-iota 5179 df-fv 5225 df-ov 5878 df-inn 8920 df-n0 9177 |
This theorem is referenced by: nn0nnaddcl 9207 elz2 9324 bcxmas 11497 |
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