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Mirrors > Home > ILE Home > Th. List > elnn | Unicode version |
Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
elnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3030 |
. . 3
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2 | sseq1 3030 |
. . 3
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3 | sseq1 3030 |
. . 3
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4 | sseq1 3030 |
. . 3
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5 | 0ss 3299 |
. . 3
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6 | unss 3157 |
. . . . . 6
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7 | vex 2613 |
. . . . . . . 8
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8 | 7 | snss 3535 |
. . . . . . 7
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9 | 8 | anbi2i 445 |
. . . . . 6
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10 | df-suc 4155 |
. . . . . . 7
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11 | 10 | sseq1i 3033 |
. . . . . 6
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12 | 6, 9, 11 | 3bitr4i 210 |
. . . . 5
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13 | 12 | biimpi 118 |
. . . 4
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14 | 13 | expcom 114 |
. . 3
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15 | 1, 2, 3, 4, 5, 14 | finds 4370 |
. 2
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16 | ssel2 3004 |
. . 3
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17 | 16 | ancoms 264 |
. 2
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18 | 15, 17 | sylan2 280 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2612 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-uni 3623 df-int 3658 df-suc 4155 df-iom 4361 |
This theorem is referenced by: ordom 4376 peano2b 4384 nndifsnid 6168 nnaordi 6169 nnmordi 6177 fidceq 6426 nnwetri 6461 |
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