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Theorem truni 4164
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 4163 . 2  |-  ( A. x  e.  A  Tr  x  ->  Tr  U_ x  e.  A  x )
2 uniiun 3992 . . 3  |-  U. A  =  U_ x  e.  A  x
3 treq 4156 . . 3  |-  ( U. A  =  U_ x  e.  A  x  ->  ( Tr  U. A  <->  Tr  U_ x  e.  A  x )
)
42, 3ax-mp 8 . 2  |-  ( Tr 
U. A  <->  Tr  U_ x  e.  A  x )
51, 4sylibr 203 1  |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633   A.wral 2577   U.cuni 3864   U_ciun 3942   Tr wtr 4150
This theorem is referenced by:  dfon2lem1  24524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-v 2824  df-in 3193  df-ss 3200  df-uni 3865  df-iun 3944  df-tr 4151
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