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Theorem triun 4126
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
triun  |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem triun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 3909 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
2 r19.29 2683 . . . . 5  |-  ( ( A. x  e.  A  Tr  B  /\  E. x  e.  A  y  e.  B )  ->  E. x  e.  A  ( Tr  B  /\  y  e.  B
) )
3 nfcv 2419 . . . . . . 7  |-  F/_ x
y
4 nfiu1 3933 . . . . . . 7  |-  F/_ x U_ x  e.  A  B
53, 4nfss 3173 . . . . . 6  |-  F/ x  y  C_  U_ x  e.  A  B
6 trss 4122 . . . . . . . 8  |-  ( Tr  B  ->  ( y  e.  B  ->  y  C_  B ) )
76imp 418 . . . . . . 7  |-  ( ( Tr  B  /\  y  e.  B )  ->  y  C_  B )
8 ssiun2 3945 . . . . . . . 8  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
9 sstr2 3186 . . . . . . . 8  |-  ( y 
C_  B  ->  ( B  C_  U_ x  e.  A  B  ->  y  C_ 
U_ x  e.  A  B ) )
108, 9syl5com 26 . . . . . . 7  |-  ( x  e.  A  ->  (
y  C_  B  ->  y 
C_  U_ x  e.  A  B ) )
117, 10syl5 28 . . . . . 6  |-  ( x  e.  A  ->  (
( Tr  B  /\  y  e.  B )  ->  y  C_  U_ x  e.  A  B ) )
125, 11rexlimi 2660 . . . . 5  |-  ( E. x  e.  A  ( Tr  B  /\  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
132, 12syl 15 . . . 4  |-  ( ( A. x  e.  A  Tr  B  /\  E. x  e.  A  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
141, 13sylan2b 461 . . 3  |-  ( ( A. x  e.  A  Tr  B  /\  y  e.  U_ x  e.  A  B )  ->  y  C_ 
U_ x  e.  A  B )
1514ralrimiva 2626 . 2  |-  ( A. x  e.  A  Tr  B  ->  A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
16 dftr3 4117 . 2  |-  ( Tr 
U_ x  e.  A  B 
<-> 
A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
1715, 16sylibr 203 1  |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U_ciun 3905   Tr wtr 4113
This theorem is referenced by:  truni  4127  r1tr  7448  r1elssi  7477
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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