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Mirrors > Home > ILE Home > Th. List > ordsuc | Unicode version |
Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.) |
Ref | Expression |
---|---|
ordsuc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucim 4424 |
. 2
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2 | en2lp 4477 |
. . . . . . . . . 10
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3 | eleq1 2203 |
. . . . . . . . . . . . 13
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4 | 3 | biimpac 296 |
. . . . . . . . . . . 12
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5 | 4 | anim2i 340 |
. . . . . . . . . . 11
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6 | 5 | expr 373 |
. . . . . . . . . 10
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7 | 2, 6 | mtoi 654 |
. . . . . . . . 9
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8 | 7 | adantl 275 |
. . . . . . . 8
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9 | elelsuc 4339 |
. . . . . . . . . . . . . . 15
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10 | 9 | adantr 274 |
. . . . . . . . . . . . . 14
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11 | ordelss 4309 |
. . . . . . . . . . . . . 14
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12 | 10, 11 | sylan2 284 |
. . . . . . . . . . . . 13
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13 | 12 | sseld 3101 |
. . . . . . . . . . . 12
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14 | 13 | expr 373 |
. . . . . . . . . . 11
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15 | 14 | pm2.43d 50 |
. . . . . . . . . 10
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16 | 15 | impr 377 |
. . . . . . . . 9
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17 | elsuci 4333 |
. . . . . . . . 9
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18 | 16, 17 | syl 14 |
. . . . . . . 8
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19 | 8, 18 | ecased 1328 |
. . . . . . 7
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20 | 19 | ancom2s 556 |
. . . . . 6
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21 | 20 | ex 114 |
. . . . 5
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22 | 21 | alrimivv 1848 |
. . . 4
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23 | dftr2 4036 |
. . . 4
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24 | 22, 23 | sylibr 133 |
. . 3
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25 | sssucid 4345 |
. . . 4
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26 | trssord 4310 |
. . . 4
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27 | 25, 26 | mp3an2 1304 |
. . 3
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28 | 24, 27 | mpancom 419 |
. 2
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29 | 1, 28 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-uni 3745 df-tr 4035 df-iord 4296 df-suc 4301 |
This theorem is referenced by: nlimsucg 4489 ordpwsucss 4490 |
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