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Theorem trint 4128
 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 4117 . . . . . 6
21ralbii 2567 . . . . 5
32biimpi 186 . . . 4
4 df-ral 2548 . . . . . 6
54ralbii 2567 . . . . 5
6 ralcom4 2806 . . . . 5
75, 6bitri 240 . . . 4
83, 7sylib 188 . . 3
9 ralim 2614 . . . 4
109alimi 1546 . . 3
118, 10syl 15 . 2
12 dftr3 4117 . . 3
13 df-ral 2548 . . . 4
14 vex 2791 . . . . . . 7
1514elint2 3869 . . . . . 6
16 ssint 3878 . . . . . 6
1715, 16imbi12i 316 . . . . 5
1817albii 1553 . . . 4
1913, 18bitri 240 . . 3
2012, 19bitri 240 . 2
2111, 20sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1527   wcel 1684  wral 2543   wss 3152  cint 3862   wtr 4113 This theorem is referenced by:  tctr  7425  intwun  8357  intgru  8436  dfon2lem8  24146 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-int 3863  df-tr 4114
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