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Theorem trint 4512
 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.)
Assertion
Ref Expression
trint
Distinct variable group:   ,

Proof of Theorem trint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dftr3 4501 . . . . 5
21ralbii 2819 . . . 4
3 df-ral 2742 . . . . . 6
43ralbii 2819 . . . . 5
5 ralcom4 3066 . . . . 5
64, 5bitri 253 . . . 4
72, 6sylbb 201 . . 3
8 ralim 2777 . . . 4
98alimi 1684 . . 3
107, 9syl 17 . 2
11 dftr3 4501 . . 3
12 df-ral 2742 . . . 4
13 vex 3048 . . . . . . 7
1413elint2 4241 . . . . . 6
15 ssint 4250 . . . . . 6
1614, 15imbi12i 328 . . . . 5
1716albii 1691 . . . 4
1812, 17bitri 253 . . 3
1911, 18bitri 253 . 2
2010, 19sylibr 216 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1442   wcel 1887  wral 2737   wss 3404  cint 4234   wtr 4497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-v 3047  df-in 3411  df-ss 3418  df-uni 4199  df-int 4235  df-tr 4498 This theorem is referenced by:  tctr  8224  intwun  9160  intgru  9239  dfon2lem8  30436
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