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Mirrors > Home > ILE Home > Th. List > nn0addcli | Unicode version |
Description: Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0addcl.1 | |
nn0addcl.2 |
Ref | Expression |
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nn0addcli |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcl.1 | . 2 | |
2 | nn0addcl.2 | . 2 | |
3 | nn0addcl 8980 | . 2 | |
4 | 1, 2, 3 | mp2an 422 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 1465 (class class class)co 5742 caddc 7591 cn0 8945 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0id 7696 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-inn 8689 df-n0 8946 |
This theorem is referenced by: numcl 9162 deccl 9164 numsucc 9189 |
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