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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 4078 | . . . . . 6 | |
2 | 1 | ralbii 2470 | . . . . 5 |
3 | 2 | biimpi 119 | . . . 4 |
4 | df-ral 2447 | . . . . . 6 | |
5 | 4 | ralbii 2470 | . . . . 5 |
6 | ralcom4 2743 | . . . . 5 | |
7 | 5, 6 | bitri 183 | . . . 4 |
8 | 3, 7 | sylib 121 | . . 3 |
9 | ralim 2523 | . . . 4 | |
10 | 9 | alimi 1442 | . . 3 |
11 | 8, 10 | syl 14 | . 2 |
12 | dftr3 4078 | . . 3 | |
13 | df-ral 2447 | . . . 4 | |
14 | vex 2724 | . . . . . . 7 | |
15 | 14 | elint2 3825 | . . . . . 6 |
16 | ssint 3834 | . . . . . 6 | |
17 | 15, 16 | imbi12i 238 | . . . . 5 |
18 | 17 | albii 1457 | . . . 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 1340 wcel 2135 wral 2442 wss 3111 cint 3818 wtr 4074 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-in 3117 df-ss 3124 df-uni 3784 df-int 3819 df-tr 4075 |
This theorem is referenced by: onintonm 4488 |
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