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Mirrors > Home > ILE Home > Th. List > triun | Unicode version |
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
triun |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 3734 |
. . . 4
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2 | r19.29 2506 |
. . . . 5
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3 | nfcv 2228 |
. . . . . . 7
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4 | nfiu1 3760 |
. . . . . . 7
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5 | 3, 4 | nfss 3018 |
. . . . . 6
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6 | trss 3945 |
. . . . . . . 8
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7 | 6 | imp 122 |
. . . . . . 7
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8 | ssiun2 3773 |
. . . . . . . 8
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9 | sstr2 3032 |
. . . . . . . 8
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10 | 8, 9 | syl5com 29 |
. . . . . . 7
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11 | 7, 10 | syl5 32 |
. . . . . 6
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12 | 5, 11 | rexlimi 2482 |
. . . . 5
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13 | 2, 12 | syl 14 |
. . . 4
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14 | 1, 13 | sylan2b 281 |
. . 3
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15 | 14 | ralrimiva 2446 |
. 2
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16 | dftr3 3940 |
. 2
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17 | 15, 16 | sylibr 132 |
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-rex 2365 df-v 2621 df-in 3005 df-ss 3012 df-uni 3654 df-iun 3732 df-tr 3937 |
This theorem is referenced by: truni 3950 |
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