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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2095 |
. . 3
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2 | eqeq1 2095 |
. . . 4
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3 | 2 | rexbidv 2382 |
. . 3
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4 | 1, 3 | orbi12d 743 |
. 2
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5 | eqeq1 2095 |
. . 3
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6 | eqeq1 2095 |
. . . 4
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7 | 6 | rexbidv 2382 |
. . 3
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8 | 5, 7 | orbi12d 743 |
. 2
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9 | eqeq1 2095 |
. . 3
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10 | eqeq1 2095 |
. . . 4
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11 | 10 | rexbidv 2382 |
. . 3
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12 | 9, 11 | orbi12d 743 |
. 2
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13 | eqeq1 2095 |
. . 3
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14 | eqeq1 2095 |
. . . 4
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15 | 14 | rexbidv 2382 |
. . 3
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16 | 13, 15 | orbi12d 743 |
. 2
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17 | eqid 2089 |
. . 3
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18 | 17 | orci 686 |
. 2
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19 | eqid 2089 |
. . . . 5
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20 | suceq 4238 |
. . . . . . 7
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21 | 20 | eqeq2d 2100 |
. . . . . 6
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22 | 21 | rspcev 2723 |
. . . . 5
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23 | 19, 22 | mpan2 417 |
. . . 4
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24 | 23 | olcd 689 |
. . 3
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25 | 24 | a1d 22 |
. 2
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26 | 4, 8, 12, 16, 18, 25 | finds 4428 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-uni 3660 df-int 3695 df-suc 4207 df-iom 4419 |
This theorem is referenced by: nnsuc 4443 nnpredcl 4449 frecabcl 6178 nnsucuniel 6270 nneneq 6627 phpm 6635 dif1enen 6650 fin0 6655 fin0or 6656 diffisn 6663 |
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