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Mirrors > Home > ILE Home > Th. List > nndceq | Unicode version |
Description: Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where is zero, see nndceq0 4577. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nndceq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri3or 6440 | . . 3 | |
2 | elirr 4500 | . . . . . . 7 | |
3 | eleq2 2221 | . . . . . . 7 | |
4 | 2, 3 | mtbii 664 | . . . . . 6 |
5 | 4 | con2i 617 | . . . . 5 |
6 | 5 | olcd 724 | . . . 4 |
7 | orc 702 | . . . 4 | |
8 | elirr 4500 | . . . . . . 7 | |
9 | eleq2 2221 | . . . . . . 7 | |
10 | 8, 9 | mtbiri 665 | . . . . . 6 |
11 | 10 | con2i 617 | . . . . 5 |
12 | 11 | olcd 724 | . . . 4 |
13 | 6, 7, 12 | 3jaoi 1285 | . . 3 |
14 | 1, 13 | syl 14 | . 2 |
15 | df-dc 821 | . 2 DECID | |
16 | 14, 15 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 820 w3o 962 wceq 1335 wcel 2128 com 4549 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-int 3808 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 |
This theorem is referenced by: nndifsnid 6454 fidceq 6814 unsnfidcex 6864 unsnfidcel 6865 enqdc 7281 nninfsellemdc 13582 |
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