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Mirrors > Home > ILE Home > Th. List > trel | Unicode version |
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
trel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 3856 | . 2 | |
2 | eleq12 2102 | . . . . . 6 | |
3 | eleq1 2100 | . . . . . . 7 | |
4 | 3 | adantl 262 | . . . . . 6 |
5 | 2, 4 | anbi12d 442 | . . . . 5 |
6 | eleq1 2100 | . . . . . 6 | |
7 | 6 | adantr 261 | . . . . 5 |
8 | 5, 7 | imbi12d 223 | . . . 4 |
9 | 8 | spc2gv 2643 | . . 3 |
10 | 9 | pm2.43b 46 | . 2 |
11 | 1, 10 | sylbi 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 wtr 3854 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 |
This theorem is referenced by: trel3 3862 trintssm 3870 ordtr1 4125 suctr 4158 trsuc 4159 ordn2lp 4269 |
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