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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 4630 |
. . . . 5
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2 | 1 | ad3antlr 493 |
. . . 4
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3 | simpr 110 |
. . . . 5
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4 | simprr 531 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | adantr 276 |
. . . . 5
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6 | 3, 5 | jca 306 |
. . . 4
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7 | ontr1 4410 |
. . . 4
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8 | 2, 6, 7 | sylc 62 |
. . 3
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9 | simpr 110 |
. . . 4
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10 | 4 | adantr 276 |
. . . 4
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11 | 9, 10 | eqeltrd 2266 |
. . 3
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12 | simplrl 535 |
. . . . 5
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13 | simpr 110 |
. . . . 5
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14 | 12, 13 | sseldd 3171 |
. . . 4
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15 | simplr 528 |
. . . . . . 7
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16 | elnn 4626 |
. . . . . . 7
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17 | 4, 15, 16 | syl2anc 411 |
. . . . . 6
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18 | 17 | adantr 276 |
. . . . 5
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19 | nnord 4632 |
. . . . 5
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20 | ordirr 4562 |
. . . . 5
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21 | 18, 19, 20 | 3syl 17 |
. . . 4
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22 | 14, 21 | pm2.21dd 621 |
. . 3
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23 | simpll 527 |
. . . 4
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24 | nntri3or 6522 |
. . . 4
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25 | 23, 17, 24 | syl2anc 411 |
. . 3
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26 | 8, 11, 22, 25 | mpjao3dan 1318 |
. 2
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27 | 26 | ex 115 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-uni 3828 df-int 3863 df-tr 4120 df-iord 4387 df-on 4389 df-suc 4392 df-iom 4611 |
This theorem is referenced by: (None) |
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