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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 2194 |
. . 3
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2 | eqeq1 2194 |
. . . 4
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3 | 2 | rexbidv 2488 |
. . 3
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4 | 1, 3 | orbi12d 794 |
. 2
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5 | eqeq1 2194 |
. . 3
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6 | eqeq1 2194 |
. . . 4
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7 | 6 | rexbidv 2488 |
. . 3
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8 | 5, 7 | orbi12d 794 |
. 2
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9 | eqeq1 2194 |
. . 3
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10 | eqeq1 2194 |
. . . 4
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11 | 10 | rexbidv 2488 |
. . 3
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12 | 9, 11 | orbi12d 794 |
. 2
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13 | eqeq1 2194 |
. . 3
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14 | eqeq1 2194 |
. . . 4
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15 | 14 | rexbidv 2488 |
. . 3
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16 | 13, 15 | orbi12d 794 |
. 2
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17 | eqid 2187 |
. . 3
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18 | 17 | orci 732 |
. 2
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19 | eqid 2187 |
. . . . 5
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20 | suceq 4414 |
. . . . . . 7
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21 | 20 | eqeq2d 2199 |
. . . . . 6
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22 | 21 | rspcev 2853 |
. . . . 5
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23 | 19, 22 | mpan2 425 |
. . . 4
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24 | 23 | olcd 735 |
. . 3
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25 | 24 | a1d 22 |
. 2
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26 | 4, 8, 12, 16, 18, 25 | finds 4611 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-suc 4383 df-iom 4602 |
This theorem is referenced by: nnsuc 4627 nnpredcl 4634 frecabcl 6414 nnsucuniel 6510 nneneq 6871 phpm 6879 dif1enen 6894 fin0 6899 fin0or 6900 diffisn 6907 |
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