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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr3 4025 | . . . . . 6 | |
2 | 1 | ralbii 2439 | . . . . 5 |
3 | 2 | biimpi 119 | . . . 4 |
4 | df-ral 2419 | . . . . . 6 | |
5 | 4 | ralbii 2439 | . . . . 5 |
6 | ralcom4 2703 | . . . . 5 | |
7 | 5, 6 | bitri 183 | . . . 4 |
8 | 3, 7 | sylib 121 | . . 3 |
9 | ralim 2489 | . . . 4 | |
10 | 9 | alimi 1431 | . . 3 |
11 | 8, 10 | syl 14 | . 2 |
12 | dftr3 4025 | . . 3 | |
13 | df-ral 2419 | . . . 4 | |
14 | vex 2684 | . . . . . . 7 | |
15 | 14 | elint2 3773 | . . . . . 6 |
16 | ssint 3782 | . . . . . 6 | |
17 | 15, 16 | imbi12i 238 | . . . . 5 |
18 | 17 | albii 1446 | . . . 4 |
19 | 13, 18 | bitri 183 | . . 3 |
20 | 12, 19 | bitri 183 | . 2 |
21 | 11, 20 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1329 wcel 1480 wral 2414 wss 3066 cint 3766 wtr 4021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-uni 3732 df-int 3767 df-tr 4022 |
This theorem is referenced by: onintonm 4428 |
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