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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 6095 | . . 3 | |
2 | orc 665 | . . . 4 | |
3 | elirr 4284 | . . . . . 6 | |
4 | eleq1 2141 | . . . . . 6 | |
5 | 3, 4 | mtbiri 632 | . . . . 5 |
6 | 5 | olcd 685 | . . . 4 |
7 | en2lp 4297 | . . . . . 6 | |
8 | 7 | imnani 657 | . . . . 5 |
9 | 8 | olcd 685 | . . . 4 |
10 | 2, 6, 9 | 3jaoi 1234 | . . 3 |
11 | 1, 10 | syl 14 | . 2 |
12 | df-dc 776 | . 2 DECID | |
13 | 11, 12 | sylibr 132 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 DECID wdc 775 w3o 918 wceq 1284 wcel 1433 com 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 |
This theorem is referenced by: ltdcpi 6513 |
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