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Mirrors > Home > MPE Home > Th. List > trel | Structured version Visualization version 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 4512 |
. 2
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2 | eleq12 2529 |
. . . . . 6
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3 | eleq1 2527 |
. . . . . . 7
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4 | 3 | adantl 472 |
. . . . . 6
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5 | 2, 4 | anbi12d 722 |
. . . . 5
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6 | eleq1 2527 |
. . . . . 6
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7 | 6 | adantr 471 |
. . . . 5
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8 | 5, 7 | imbi12d 326 |
. . . 4
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9 | 8 | spc2gv 3148 |
. . 3
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10 | 9 | pm2.43b 52 |
. 2
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11 | 1, 10 | sylbi 200 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 |
This theorem depends on definitions: df-bi 190 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-v 3058 df-in 3422 df-ss 3429 df-uni 4212 df-tr 4511 |
This theorem is referenced by: trel3 4518 trintss 4526 ordn2lp 5461 ordelord 5463 tz7.7 5467 ordtr1 5484 suctr 5524 trsuc 5525 ordom 6727 elnn 6728 epfrs 8240 tcrank 8380 dfon2lem6 30482 tratrb 36940 truniALT 36945 onfrALTlem2 36955 trelded 36975 pwtrrVD 37260 suctrALT 37261 suctrALT2VD 37271 suctrALT2 37272 tratrbVD 37297 truniALTVD 37314 trintALTVD 37316 trintALT 37317 onfrALTlem2VD 37325 suctrALTcf 37358 suctrALTcfVD 37359 |
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