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Mirrors > Home > ILE Home > Th. List > dmaddpi | Unicode version |
Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
dmaddpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4632 | . . 3 | |
2 | fnoa 6027 | . . . . 5 | |
3 | fndm 4998 | . . . . 5 | |
4 | 2, 3 | ax-mp 7 | . . . 4 |
5 | 4 | ineq2i 3135 | . . 3 |
6 | 1, 5 | eqtri 2060 | . 2 |
7 | df-pli 6403 | . . 3 | |
8 | 7 | dmeqi 4536 | . 2 |
9 | df-ni 6402 | . . . . . . 7 | |
10 | difss 3070 | . . . . . . 7 | |
11 | 9, 10 | eqsstri 2975 | . . . . . 6 |
12 | omsson 4335 | . . . . . 6 | |
13 | 11, 12 | sstri 2954 | . . . . 5 |
14 | anidm 376 | . . . . 5 | |
15 | 13, 14 | mpbir 134 | . . . 4 |
16 | xpss12 4445 | . . . 4 | |
17 | 15, 16 | ax-mp 7 | . . 3 |
18 | dfss 2932 | . . 3 | |
19 | 17, 18 | mpbi 133 | . 2 |
20 | 6, 8, 19 | 3eqtr4i 2070 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wceq 1243 cdif 2914 cin 2916 wss 2917 c0 3224 csn 3375 con0 4100 com 4313 cxp 4343 cdm 4345 cres 4347 wfn 4897 coa 5998 cnpi 6370 cpli 6371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-ni 6402 df-pli 6403 |
This theorem is referenced by: (None) |
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