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Mirrors > Home > ILE Home > Th. List > addclpi | Unicode version |
Description: Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
addclpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addpiord 7117 | . 2 | |
2 | pinn 7110 | . . 3 | |
3 | pinn 7110 | . . . . 5 | |
4 | nnacl 6369 | . . . . 5 | |
5 | 3, 4 | sylan2 284 | . . . 4 |
6 | elni2 7115 | . . . . 5 | |
7 | nnaordi 6397 | . . . . . . . 8 | |
8 | ne0i 3364 | . . . . . . . 8 | |
9 | 7, 8 | syl6 33 | . . . . . . 7 |
10 | 9 | expcom 115 | . . . . . 6 |
11 | 10 | imp32 255 | . . . . 5 |
12 | 6, 11 | sylan2b 285 | . . . 4 |
13 | elni 7109 | . . . 4 | |
14 | 5, 12, 13 | sylanbrc 413 | . . 3 |
15 | 2, 14 | sylan 281 | . 2 |
16 | 1, 15 | eqeltrd 2214 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 wne 2306 c0 3358 com 4499 (class class class)co 5767 coa 6303 cnpi 7073 cpli 7074 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-ni 7105 df-pli 7106 |
This theorem is referenced by: addasspig 7131 distrpig 7134 ltapig 7139 1lt2pi 7141 indpi 7143 addcmpblnq 7168 addpipqqslem 7170 addclnq 7176 addassnqg 7183 distrnqg 7188 ltanqg 7201 1lt2nq 7207 ltexnqq 7209 archnqq 7218 prarloclemarch2 7220 nqnq0a 7255 nntopi 7695 |
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