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Mirrors > Home > ILE Home > Th. List > addnidpig | Unicode version |
Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
Ref | Expression |
---|---|
addnidpig |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6922 |
. . 3
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2 | elni2 6927 |
. . . 4
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3 | nnaordi 6281 |
. . . . . . 7
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4 | nna0 6249 |
. . . . . . . . . 10
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5 | 4 | eleq1d 2157 |
. . . . . . . . 9
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6 | nnord 4439 |
. . . . . . . . . . . 12
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7 | ordirr 4371 |
. . . . . . . . . . . 12
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8 | 6, 7 | syl 14 |
. . . . . . . . . . 11
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9 | eleq2 2152 |
. . . . . . . . . . . 12
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10 | 9 | notbid 628 |
. . . . . . . . . . 11
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11 | 8, 10 | syl5ibrcom 156 |
. . . . . . . . . 10
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12 | 11 | con2d 590 |
. . . . . . . . 9
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13 | 5, 12 | sylbid 149 |
. . . . . . . 8
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14 | 13 | adantl 272 |
. . . . . . 7
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15 | 3, 14 | syld 45 |
. . . . . 6
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16 | 15 | expcom 115 |
. . . . 5
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17 | 16 | imp32 254 |
. . . 4
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18 | 2, 17 | sylan2b 282 |
. . 3
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19 | 1, 18 | sylan 278 |
. 2
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20 | addpiord 6929 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | eqeq1d 2097 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 19, 21 | mtbird 634 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-oadd 6199 df-ni 6917 df-pli 6918 |
This theorem is referenced by: (None) |
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