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Theorem triun 4249
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 4032 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
2 r19.29 2782 . . . . 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 2516 . . . . . . 7  |-  F/_ x
y
4 nfiu1 4056 . . . . . . 7  |-  F/_ x U_ x  e.  A  B
53, 4nfss 3277 . . . . . 6  |-  F/ x  y  C_  U_ x  e.  A  B
6 trss 4245 . . . . . . . 8  |-  ( Tr  B  ->  ( y  e.  B  ->  y  C_  B ) )
76imp 419 . . . . . . 7  |-  ( ( Tr  B  /\  y  e.  B )  ->  y  C_  B )
8 ssiun2 4068 . . . . . . . 8  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
9 sstr2 3291 . . . . . . . 8  |-  ( y 
C_  B  ->  ( B  C_  U_ x  e.  A  B  ->  y  C_ 
U_ x  e.  A  B ) )
108, 9syl5com 28 . . . . . . 7  |-  ( x  e.  A  ->  (
y  C_  B  ->  y 
C_  U_ x  e.  A  B ) )
117, 10syl5 30 . . . . . 6  |-  ( x  e.  A  ->  (
( Tr  B  /\  y  e.  B )  ->  y  C_  U_ x  e.  A  B ) )
125, 11rexlimi 2759 . . . . 5  |-  ( E. x  e.  A  ( Tr  B  /\  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
132, 12syl 16 . . . 4  |-  ( ( A. x  e.  A  Tr  B  /\  E. x  e.  A  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
141, 13sylan2b 462 . . 3  |-  ( ( A. x  e.  A  Tr  B  /\  y  e.  U_ x  e.  A  B )  ->  y  C_ 
U_ x  e.  A  B )
1514ralrimiva 2725 . 2  |-  ( A. x  e.  A  Tr  B  ->  A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
16 dftr3 4240 . 2  |-  ( Tr 
U_ x  e.  A  B 
<-> 
A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
1715, 16sylibr 204 1  |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   U_ciun 4028   Tr wtr 4236
This theorem is referenced by:  truni  4250  r1tr  7628  r1elssi  7657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894  df-in 3263  df-ss 3270  df-uni 3951  df-iun 4030  df-tr 4237
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