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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 2124 | . . 3 | |
2 | eqeq1 2124 | . . . 4 | |
3 | 2 | rexbidv 2415 | . . 3 |
4 | 1, 3 | orbi12d 767 | . 2 |
5 | eqeq1 2124 | . . 3 | |
6 | eqeq1 2124 | . . . 4 | |
7 | 6 | rexbidv 2415 | . . 3 |
8 | 5, 7 | orbi12d 767 | . 2 |
9 | eqeq1 2124 | . . 3 | |
10 | eqeq1 2124 | . . . 4 | |
11 | 10 | rexbidv 2415 | . . 3 |
12 | 9, 11 | orbi12d 767 | . 2 |
13 | eqeq1 2124 | . . 3 | |
14 | eqeq1 2124 | . . . 4 | |
15 | 14 | rexbidv 2415 | . . 3 |
16 | 13, 15 | orbi12d 767 | . 2 |
17 | eqid 2117 | . . 3 | |
18 | 17 | orci 705 | . 2 |
19 | eqid 2117 | . . . . 5 | |
20 | suceq 4294 | . . . . . . 7 | |
21 | 20 | eqeq2d 2129 | . . . . . 6 |
22 | 21 | rspcev 2763 | . . . . 5 |
23 | 19, 22 | mpan2 421 | . . . 4 |
24 | 23 | olcd 708 | . . 3 |
25 | 24 | a1d 22 | . 2 |
26 | 4, 8, 12, 16, 18, 25 | finds 4484 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 682 wceq 1316 wcel 1465 wrex 2394 c0 3333 csuc 4257 com 4474 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 |
This theorem is referenced by: nnsuc 4499 nnpredcl 4506 frecabcl 6264 nnsucuniel 6359 nneneq 6719 phpm 6727 dif1enen 6742 fin0 6747 fin0or 6748 diffisn 6755 |
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