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Mirrors > Home > ILE Home > Th. List > nndcel | Unicode version |
Description: Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.) |
Ref | Expression |
---|---|
nndcel | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6072 | . . 3 | |
2 | orc 633 | . . . 4 | |
3 | elirr 4266 | . . . . . 6 | |
4 | eleq1 2100 | . . . . . 6 | |
5 | 3, 4 | mtbiri 600 | . . . . 5 |
6 | 5 | olcd 653 | . . . 4 |
7 | en2lp 4278 | . . . . . 6 | |
8 | 7 | imnani 625 | . . . . 5 |
9 | 8 | olcd 653 | . . . 4 |
10 | 2, 6, 9 | 3jaoi 1198 | . . 3 |
11 | 1, 10 | syl 14 | . 2 |
12 | df-dc 743 | . 2 DECID | |
13 | 11, 12 | sylibr 137 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 DECID wdc 742 w3o 884 wceq 1243 wcel 1393 com 4313 |
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-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-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 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-uni 3581 df-int 3616 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 |
This theorem is referenced by: ltdcpi 6421 |
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