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Theorem truni 4101
Description: The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
Assertion
Ref Expression
truni  |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
Distinct variable group:    x, A

Proof of Theorem truni
StepHypRef Expression
1 triun 4100 . 2  |-  ( A. x  e.  A  Tr  x  ->  Tr  U_ x  e.  A  x )
2 uniiun 3929 . . 3  |-  U. A  =  U_ x  e.  A  x
3 treq 4093 . . 3  |-  ( U. A  =  U_ x  e.  A  x  ->  ( Tr  U. A  <->  Tr  U_ x  e.  A  x )
)
42, 3ax-mp 10 . 2  |-  ( Tr 
U. A  <->  Tr  U_ x  e.  A  x )
51, 4sylibr 205 1  |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619   A.wral 2518   U.cuni 3801   U_ciun 3879   Tr wtr 4087
This theorem is referenced by:  dfon2lem1  23508
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-v 2765  df-in 3134  df-ss 3141  df-uni 3802  df-iun 3881  df-tr 4088
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