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Mirrors > Home > ILE Home > Th. List > nn0addcl | Unicode version |
Description: Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
nn0addcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 8427 |
. 2
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2 | id 19 |
. . 3
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3 | df-n0 8674 |
. . 3
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4 | nnaddcl 8442 |
. . . 4
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5 | 4 | adantl 271 |
. . 3
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6 | 2, 3, 5 | un0addcl 8706 |
. 2
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7 | 1, 6 | mpan 415 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-addcom 7445 ax-addass 7447 ax-i2m1 7450 ax-0id 7453 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 df-inn 8423 df-n0 8674 |
This theorem is referenced by: nn0addcli 8710 peano2nn0 8713 nn0addcld 8730 nn0readdcl 8732 difelfznle 9546 elfzodifsumelfzo 9612 expadd 9997 faclbnd6 10152 facavg 10154 fsumnn0cl 10797 bcxmas 10883 eftlub 10980 |
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