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Mirrors > Home > ILE Home > Th. List > nntr2 | Unicode version |
Description: Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.) |
Ref | Expression |
---|---|
nntr2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4483 |
. . . . 5
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2 | 1 | ad3antlr 482 |
. . . 4
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3 | simpr 109 |
. . . . 5
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4 | simprr 504 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | adantr 272 |
. . . . 5
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6 | 3, 5 | jca 302 |
. . . 4
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7 | ontr1 4271 |
. . . 4
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8 | 2, 6, 7 | sylc 62 |
. . 3
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9 | simpr 109 |
. . . 4
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10 | 4 | adantr 272 |
. . . 4
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11 | 9, 10 | eqeltrd 2191 |
. . 3
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12 | simplrl 507 |
. . . . 5
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13 | simpr 109 |
. . . . 5
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14 | 12, 13 | sseldd 3064 |
. . . 4
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15 | simplr 502 |
. . . . . . 7
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16 | elnn 4479 |
. . . . . . 7
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17 | 4, 15, 16 | syl2anc 406 |
. . . . . 6
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18 | 17 | adantr 272 |
. . . . 5
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19 | nnord 4485 |
. . . . 5
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20 | ordirr 4417 |
. . . . 5
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21 | 18, 19, 20 | 3syl 17 |
. . . 4
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22 | 14, 21 | pm2.21dd 592 |
. . 3
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23 | simpll 501 |
. . . 4
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24 | nntri3or 6343 |
. . . 4
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25 | 23, 17, 24 | syl2anc 406 |
. . 3
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26 | 8, 11, 22, 25 | mpjao3dan 1268 |
. 2
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27 | 26 | ex 114 |
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-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-3or 946 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-ne 2283 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-tr 3987 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 |
This theorem is referenced by: (None) |
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