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Theorem trint 3897
 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 3886 . . . . . 6
21ralbii 2347 . . . . 5
32biimpi 117 . . . 4
4 df-ral 2328 . . . . . 6
54ralbii 2347 . . . . 5
6 ralcom4 2593 . . . . 5
75, 6bitri 177 . . . 4
83, 7sylib 131 . . 3
9 ralim 2397 . . . 4
109alimi 1360 . . 3
118, 10syl 14 . 2
12 dftr3 3886 . . 3
13 df-ral 2328 . . . 4
14 vex 2577 . . . . . . 7
1514elint2 3650 . . . . . 6
16 ssint 3659 . . . . . 6
1715, 16imbi12i 232 . . . . 5
1817albii 1375 . . . 4
1913, 18bitri 177 . . 3
2012, 19bitri 177 . 2
2111, 20sylibr 141 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1257   wcel 1409  wral 2323   wss 2945  cint 3643   wtr 3882 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-int 3644  df-tr 3883 This theorem is referenced by:  onintonm  4271
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