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Theorem truni 4229
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 4228 . 2  |-  ( A. x  e.  A  Tr  x  ->  Tr  U_ x  e.  A  x )
2 uniiun 4057 . . 3  |-  U. A  =  U_ x  e.  A  x
3 treq 4221 . . 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 1647   A.wral 2628   U.cuni 3929   U_ciun 4007   Tr wtr 4215
This theorem is referenced by:  dfon2lem1  24880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-v 2875  df-in 3245  df-ss 3252  df-uni 3930  df-iun 4009  df-tr 4216
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