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Mirrors > Home > ILE Home > Th. List > ordtri2or2exmidlem | Unicode version |
Description: A set which is if or if is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Ref | Expression |
---|---|
ordtri2or2exmidlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 495 | . . . . . . 7 | |
2 | noel 3255 | . . . . . . . . 9 | |
3 | eleq2 2142 | . . . . . . . . 9 | |
4 | 2, 3 | mtbiri 632 | . . . . . . . 8 |
5 | 4 | adantl 271 | . . . . . . 7 |
6 | 1, 5 | pm2.21dd 582 | . . . . . 6 |
7 | eleq2 2142 | . . . . . . . . . . 11 | |
8 | 7 | biimpac 292 | . . . . . . . . . 10 |
9 | velsn 3415 | . . . . . . . . . 10 | |
10 | 8, 9 | sylib 120 | . . . . . . . . 9 |
11 | orc 665 | . . . . . . . . . 10 | |
12 | vex 2604 | . . . . . . . . . . 11 | |
13 | 12 | elpr 3419 | . . . . . . . . . 10 |
14 | 11, 13 | sylibr 132 | . . . . . . . . 9 |
15 | 10, 14 | syl 14 | . . . . . . . 8 |
16 | 15 | adantlr 460 | . . . . . . 7 |
17 | biidd 170 | . . . . . . . . . 10 | |
18 | 17 | elrab 2749 | . . . . . . . . 9 |
19 | 18 | simprbi 269 | . . . . . . . 8 |
20 | 19 | ad2antlr 472 | . . . . . . 7 |
21 | biidd 170 | . . . . . . . 8 | |
22 | 21 | elrab 2749 | . . . . . . 7 |
23 | 16, 20, 22 | sylanbrc 408 | . . . . . 6 |
24 | elrabi 2746 | . . . . . . . 8 | |
25 | vex 2604 | . . . . . . . . 9 | |
26 | 25 | elpr 3419 | . . . . . . . 8 |
27 | 24, 26 | sylib 120 | . . . . . . 7 |
28 | 27 | adantl 271 | . . . . . 6 |
29 | 6, 23, 28 | mpjaodan 744 | . . . . 5 |
30 | 29 | gen2 1379 | . . . 4 |
31 | dftr2 3877 | . . . 4 | |
32 | 30, 31 | mpbir 144 | . . 3 |
33 | ssrab2 3079 | . . 3 | |
34 | 2ordpr 4267 | . . 3 | |
35 | trssord 4135 | . . 3 | |
36 | 32, 33, 34, 35 | mp3an 1268 | . 2 |
37 | pp0ex 3960 | . . . 4 | |
38 | 37 | rabex 3922 | . . 3 |
39 | 38 | elon 4129 | . 2 |
40 | 36, 39 | mpbir 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 wal 1282 wceq 1284 wcel 1433 crab 2352 wss 2973 c0 3251 csn 3398 cpr 3399 wtr 3875 word 4117 con0 4118 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: ordtri2or2exmid 4314 |
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