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Theorem triun 4503
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 4274 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
2 r19.29 2912 . . . . 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 2612 . . . . . . 7  |-  F/_ x
y
4 nfiu1 4299 . . . . . . 7  |-  F/_ x U_ x  e.  A  B
53, 4nfss 3411 . . . . . 6  |-  F/ x  y  C_  U_ x  e.  A  B
6 trss 4499 . . . . . . . 8  |-  ( Tr  B  ->  ( y  e.  B  ->  y  C_  B ) )
76imp 436 . . . . . . 7  |-  ( ( Tr  B  /\  y  e.  B )  ->  y  C_  B )
8 ssiun2 4312 . . . . . . . 8  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
9 sstr2 3425 . . . . . . . 8  |-  ( y 
C_  B  ->  ( B  C_  U_ x  e.  A  B  ->  y  C_ 
U_ x  e.  A  B ) )
108, 9syl5com 30 . . . . . . 7  |-  ( x  e.  A  ->  (
y  C_  B  ->  y 
C_  U_ x  e.  A  B ) )
117, 10syl5 32 . . . . . 6  |-  ( x  e.  A  ->  (
( Tr  B  /\  y  e.  B )  ->  y  C_  U_ x  e.  A  B ) )
125, 11rexlimi 2864 . . . . 5  |-  ( E. x  e.  A  ( Tr  B  /\  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
132, 12syl 17 . . . 4  |-  ( ( A. x  e.  A  Tr  B  /\  E. x  e.  A  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
141, 13sylan2b 483 . . 3  |-  ( ( A. x  e.  A  Tr  B  /\  y  e.  U_ x  e.  A  B )  ->  y  C_ 
U_ x  e.  A  B )
1514ralrimiva 2809 . 2  |-  ( A. x  e.  A  Tr  B  ->  A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
16 dftr3 4494 . 2  |-  ( Tr 
U_ x  e.  A  B 
<-> 
A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
1715, 16sylibr 217 1  |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   A.wral 2756   E.wrex 2757    C_ wss 3390   U_ciun 4269   Tr wtr 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-iun 4271  df-tr 4491
This theorem is referenced by:  truni  4504  r1tr  8265  r1elssi  8294
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