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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 2177 | . . 3 | |
2 | eqeq1 2177 | . . . 4 | |
3 | 2 | rexbidv 2471 | . . 3 |
4 | 1, 3 | orbi12d 788 | . 2 |
5 | eqeq1 2177 | . . 3 | |
6 | eqeq1 2177 | . . . 4 | |
7 | 6 | rexbidv 2471 | . . 3 |
8 | 5, 7 | orbi12d 788 | . 2 |
9 | eqeq1 2177 | . . 3 | |
10 | eqeq1 2177 | . . . 4 | |
11 | 10 | rexbidv 2471 | . . 3 |
12 | 9, 11 | orbi12d 788 | . 2 |
13 | eqeq1 2177 | . . 3 | |
14 | eqeq1 2177 | . . . 4 | |
15 | 14 | rexbidv 2471 | . . 3 |
16 | 13, 15 | orbi12d 788 | . 2 |
17 | eqid 2170 | . . 3 | |
18 | 17 | orci 726 | . 2 |
19 | eqid 2170 | . . . . 5 | |
20 | suceq 4385 | . . . . . . 7 | |
21 | 20 | eqeq2d 2182 | . . . . . 6 |
22 | 21 | rspcev 2834 | . . . . 5 |
23 | 19, 22 | mpan2 423 | . . . 4 |
24 | 23 | olcd 729 | . . 3 |
25 | 24 | a1d 22 | . 2 |
26 | 4, 8, 12, 16, 18, 25 | finds 4582 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 703 wceq 1348 wcel 2141 wrex 2449 c0 3414 csuc 4348 com 4572 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-int 3830 df-suc 4354 df-iom 4573 |
This theorem is referenced by: nnsuc 4598 nnpredcl 4605 frecabcl 6375 nnsucuniel 6471 nneneq 6831 phpm 6839 dif1enen 6854 fin0 6859 fin0or 6860 diffisn 6867 |
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