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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 3086 |
. . 3
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2 | sseq1 3086 |
. . 3
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3 | sseq1 3086 |
. . 3
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4 | sseq1 3086 |
. . 3
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5 | 0ss 3367 |
. . 3
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6 | unss 3216 |
. . . . . 6
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7 | vex 2660 |
. . . . . . . 8
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8 | 7 | snss 3615 |
. . . . . . 7
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9 | 8 | anbi2i 450 |
. . . . . 6
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10 | df-suc 4253 |
. . . . . . 7
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11 | 10 | sseq1i 3089 |
. . . . . 6
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12 | 6, 9, 11 | 3bitr4i 211 |
. . . . 5
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13 | 12 | biimpi 119 |
. . . 4
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14 | 13 | expcom 115 |
. . 3
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15 | 1, 2, 3, 4, 5, 14 | finds 4474 |
. 2
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16 | ssel2 3058 |
. . 3
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17 | 16 | ancoms 266 |
. 2
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18 | 15, 17 | sylan2 282 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-uni 3703 df-int 3738 df-suc 4253 df-iom 4465 |
This theorem is referenced by: ordom 4480 peano2b 4488 nntr2 6353 nndifsnid 6357 nnaordi 6358 nnmordi 6366 fidceq 6716 nnwetri 6757 enumctlemm 6951 ennnfonelemdm 11778 ennnfonelemnn0 11780 nnti 12883 nninfsellemdc 12898 nninfsellemeq 12902 nninfsellemeqinf 12904 |
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