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Mirrors > Home > ILE Home > Th. List > trint | Unicode version |
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) |
Ref | Expression |
---|---|
trint |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr3 3990 |
. . . . . 6
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2 | 1 | ralbii 2415 |
. . . . 5
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3 | 2 | biimpi 119 |
. . . 4
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4 | df-ral 2395 |
. . . . . 6
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5 | 4 | ralbii 2415 |
. . . . 5
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6 | ralcom4 2679 |
. . . . 5
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7 | 5, 6 | bitri 183 |
. . . 4
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8 | 3, 7 | sylib 121 |
. . 3
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9 | ralim 2465 |
. . . 4
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10 | 9 | alimi 1414 |
. . 3
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11 | 8, 10 | syl 14 |
. 2
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12 | dftr3 3990 |
. . 3
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13 | df-ral 2395 |
. . . 4
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14 | vex 2660 |
. . . . . . 7
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15 | 14 | elint2 3744 |
. . . . . 6
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16 | ssint 3753 |
. . . . . 6
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17 | 15, 16 | imbi12i 238 |
. . . . 5
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18 | 17 | albii 1429 |
. . . 4
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19 | 13, 18 | bitri 183 |
. . 3
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20 | 12, 19 | bitri 183 |
. 2
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21 | 11, 20 | 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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-in 3043 df-ss 3050 df-uni 3703 df-int 3738 df-tr 3987 |
This theorem is referenced by: onintonm 4393 |
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