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Theorem trint 3860
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  Tr  Tr  |^|
Distinct variable group:   ,

Proof of Theorem trint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dftr3 3849 . . . . . 6  Tr  C_
21ralbii 2324 . . . . 5  Tr 
C_
32biimpi 113 . . . 4  Tr  C_
4 df-ral 2305 . . . . . 6  C_  C_
54ralbii 2324 . . . . 5  C_  C_
6 ralcom4 2570 . . . . 5  C_  C_
75, 6bitri 173 . . . 4  C_  C_
83, 7sylib 127 . . 3  Tr  C_
9 ralim 2374 . . . 4  C_ 
C_
109alimi 1341 . . 3  C_  C_
118, 10syl 14 . 2  Tr 
C_
12 dftr3 3849 . . 3  Tr 
|^|  |^|  C_  |^|
13 df-ral 2305 . . . 4  |^| 
C_  |^|  |^|  C_  |^|
14 vex 2554 . . . . . . 7 
_V
1514elint2 3613 . . . . . 6  |^|
16 ssint 3622 . . . . . 6 
C_  |^|  C_
1715, 16imbi12i 228 . . . . 5  |^|  C_  |^|  C_
1817albii 1356 . . . 4  |^|  C_  |^|  C_
1913, 18bitri 173 . . 3  |^| 
C_  |^| 
C_
2012, 19bitri 173 . 2  Tr 
|^| 
C_
2111, 20sylibr 137 1  Tr  Tr  |^|
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240   wcel 1390  wral 2300    C_ wss 2911   |^|cint 3606   Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-int 3607  df-tr 3846
This theorem is referenced by: (None)
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