Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  trint Unicode version

Theorem trint 4025
 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
StepHypRef Expression
1 dftr3 4014 . . . . . 6
21ralbii 2531 . . . . 5
32biimpi 188 . . . 4
4 df-ral 2513 . . . . . 6
54ralbii 2531 . . . . 5
6 ralcom4 2744 . . . . 5
75, 6bitri 242 . . . 4
83, 7sylib 190 . . 3
9 ralim 2576 . . . 4
109alimi 1546 . . 3
118, 10syl 17 . 2
12 dftr3 4014 . . 3
13 df-ral 2513 . . . 4
14 vex 2730 . . . . . . 7
1514elint2 3767 . . . . . 6
16 ssint 3776 . . . . . 6
1715, 16imbi12i 318 . . . . 5
1817albii 1554 . . . 4
1913, 18bitri 242 . . 3
2012, 19bitri 242 . 2
2111, 20sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 6  wal 1532   wcel 1621  wral 2509   wss 3078  cint 3760   wtr 4010 This theorem is referenced by:  tctr  7309  intwun  8237  intgru  8316  dfon2lem8  23314 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729  df-in 3085  df-ss 3089  df-uni 3728  df-int 3761  df-tr 4011
 Copyright terms: Public domain W3C validator