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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 4120 |
. . . . . 6
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2 | 1 | ralbii 2496 |
. . . . 5
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3 | 2 | biimpi 120 |
. . . 4
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4 | df-ral 2473 |
. . . . . 6
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5 | 4 | ralbii 2496 |
. . . . 5
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6 | ralcom4 2774 |
. . . . 5
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7 | 5, 6 | bitri 184 |
. . . 4
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8 | 3, 7 | sylib 122 |
. . 3
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9 | ralim 2549 |
. . . 4
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10 | 9 | alimi 1466 |
. . 3
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11 | 8, 10 | syl 14 |
. 2
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12 | dftr3 4120 |
. . 3
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13 | df-ral 2473 |
. . . 4
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14 | vex 2755 |
. . . . . . 7
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15 | 14 | elint2 3866 |
. . . . . 6
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16 | ssint 3875 |
. . . . . 6
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17 | 15, 16 | imbi12i 239 |
. . . . 5
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18 | 17 | albii 1481 |
. . . 4
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19 | 13, 18 | bitri 184 |
. . 3
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20 | 12, 19 | bitri 184 |
. 2
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21 | 11, 20 | 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-v 2754 df-in 3150 df-ss 3157 df-uni 3825 df-int 3860 df-tr 4117 |
This theorem is referenced by: onintonm 4534 |
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