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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 3756 |
. . . 4
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2 | r19.29 2520 |
. . . . 5
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3 | nfcv 2235 |
. . . . . . 7
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4 | nfiu1 3782 |
. . . . . . 7
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5 | 3, 4 | nfss 3032 |
. . . . . 6
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6 | trss 3967 |
. . . . . . . 8
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7 | 6 | imp 123 |
. . . . . . 7
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8 | ssiun2 3795 |
. . . . . . . 8
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9 | sstr2 3046 |
. . . . . . . 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 2495 |
. . . . 5
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13 | 2, 12 | syl 14 |
. . . 4
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14 | 1, 13 | sylan2b 282 |
. . 3
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15 | 14 | ralrimiva 2458 |
. 2
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16 | dftr3 3962 |
. 2
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17 | 15, 16 | sylibr 133 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-in 3019 df-ss 3026 df-uni 3676 df-iun 3754 df-tr 3959 |
This theorem is referenced by: truni 3972 |
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