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Mirrors > Home > ILE Home > Th. List > nn0suc | Unicode version |
Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
nn0suc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2046 | . . 3 | |
2 | eqeq1 2046 | . . . 4 | |
3 | 2 | rexbidv 2327 | . . 3 |
4 | 1, 3 | orbi12d 707 | . 2 |
5 | eqeq1 2046 | . . 3 | |
6 | eqeq1 2046 | . . . 4 | |
7 | 6 | rexbidv 2327 | . . 3 |
8 | 5, 7 | orbi12d 707 | . 2 |
9 | eqeq1 2046 | . . 3 | |
10 | eqeq1 2046 | . . . 4 | |
11 | 10 | rexbidv 2327 | . . 3 |
12 | 9, 11 | orbi12d 707 | . 2 |
13 | eqeq1 2046 | . . 3 | |
14 | eqeq1 2046 | . . . 4 | |
15 | 14 | rexbidv 2327 | . . 3 |
16 | 13, 15 | orbi12d 707 | . 2 |
17 | eqid 2040 | . . 3 | |
18 | 17 | orci 650 | . 2 |
19 | eqid 2040 | . . . . 5 | |
20 | suceq 4139 | . . . . . . 7 | |
21 | 20 | eqeq2d 2051 | . . . . . 6 |
22 | 21 | rspcev 2656 | . . . . 5 |
23 | 19, 22 | mpan2 401 | . . . 4 |
24 | 23 | olcd 653 | . . 3 |
25 | 24 | a1d 22 | . 2 |
26 | 4, 8, 12, 16, 18, 25 | finds 4323 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 629 wceq 1243 wcel 1393 wrex 2307 c0 3224 csuc 4102 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-iinf 4311 |
This theorem depends on definitions: df-bi 110 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-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-suc 4108 df-iom 4314 |
This theorem is referenced by: nnsuc 4338 nneneq 6320 phpm 6327 fin0 6342 fin0or 6343 diffisn 6350 |
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