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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 3905 |
. . . 4
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2 | r19.29 2627 |
. . . . 5
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3 | nfcv 2332 |
. . . . . . 7
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4 | nfiu1 3931 |
. . . . . . 7
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5 | 3, 4 | nfss 3163 |
. . . . . 6
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6 | trss 4125 |
. . . . . . . 8
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7 | 6 | imp 124 |
. . . . . . 7
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8 | ssiun2 3944 |
. . . . . . . 8
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9 | sstr2 3177 |
. . . . . . . 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 2600 |
. . . . 5
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13 | 2, 12 | syl 14 |
. . . 4
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14 | 1, 13 | sylan2b 287 |
. . 3
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15 | 14 | ralrimiva 2563 |
. 2
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16 | dftr3 4120 |
. 2
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17 | 15, 16 | sylibr 134 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 df-uni 3825 df-iun 3903 df-tr 4117 |
This theorem is referenced by: truni 4130 |
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