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Mirrors > Home > ILE Home > Th. List > trss | Unicode version |
Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
trss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2141 | . . . . 5 | |
2 | sseq1 3020 | . . . . 5 | |
3 | 1, 2 | imbi12d 232 | . . . 4 |
4 | 3 | imbi2d 228 | . . 3 |
5 | dftr3 3879 | . . . 4 | |
6 | rsp 2411 | . . . 4 | |
7 | 5, 6 | sylbi 119 | . . 3 |
8 | 4, 7 | vtoclg 2658 | . 2 |
9 | 8 | pm2.43b 51 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 wcel 1433 wral 2348 wss 2973 wtr 3875 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 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-v 2603 df-in 2979 df-ss 2986 df-uni 3602 df-tr 3876 |
This theorem is referenced by: trin 3885 triun 3888 trintssm 3891 tz7.2 4109 ordelss 4134 trsucss 4178 ordsucss 4248 |
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