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Theorem truni 4127
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 4126 . 2  |-  ( A. x  e.  A  Tr  x  ->  Tr  U_ x  e.  A  x )
2 uniiun 3955 . . 3  |-  U. A  =  U_ x  e.  A  x
3 treq 4119 . . 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 1623   A.wral 2543   U.cuni 3827   U_ciun 3905   Tr wtr 4113
This theorem is referenced by:  dfon2lem1  24139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-iun 3907  df-tr 4114
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