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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2150 |
. . . . 5
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2 | sseq1 3047 |
. . . . 5
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3 | 1, 2 | imbi12d 232 |
. . . 4
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4 | 3 | imbi2d 228 |
. . 3
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5 | dftr3 3940 |
. . . 4
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6 | rsp 2423 |
. . . 4
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7 | 5, 6 | sylbi 119 |
. . 3
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8 | 4, 7 | vtoclg 2679 |
. 2
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9 | 8 | pm2.43b 51 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-in 3005 df-ss 3012 df-uni 3654 df-tr 3937 |
This theorem is referenced by: trin 3946 triun 3949 trintssm 3952 tz7.2 4181 ordelss 4206 trsucss 4250 ordsucss 4321 |
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