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Theorem triun 4098
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 3875 . . . 4  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
2 r19.29 2607 . . . . 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 2312 . . . . . . 7  |-  F/_ x
y
4 nfiu1 3901 . . . . . . 7  |-  F/_ x U_ x  e.  A  B
53, 4nfss 3140 . . . . . 6  |-  F/ x  y  C_  U_ x  e.  A  B
6 trss 4094 . . . . . . . 8  |-  ( Tr  B  ->  ( y  e.  B  ->  y  C_  B ) )
76imp 123 . . . . . . 7  |-  ( ( Tr  B  /\  y  e.  B )  ->  y  C_  B )
8 ssiun2 3914 . . . . . . . 8  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
9 sstr2 3154 . . . . . . . 8  |-  ( y 
C_  B  ->  ( B  C_  U_ x  e.  A  B  ->  y  C_ 
U_ x  e.  A  B ) )
108, 9syl5com 29 . . . . . . 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 2580 . . . . 5  |-  ( E. x  e.  A  ( Tr  B  /\  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
132, 12syl 14 . . . 4  |-  ( ( A. x  e.  A  Tr  B  /\  E. x  e.  A  y  e.  B )  ->  y  C_ 
U_ x  e.  A  B )
141, 13sylan2b 285 . . 3  |-  ( ( A. x  e.  A  Tr  B  /\  y  e.  U_ x  e.  A  B )  ->  y  C_ 
U_ x  e.  A  B )
1514ralrimiva 2543 . 2  |-  ( A. x  e.  A  Tr  B  ->  A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
16 dftr3 4089 . 2  |-  ( Tr 
U_ x  e.  A  B 
<-> 
A. y  e.  U_  x  e.  A  B
y  C_  U_ x  e.  A  B )
1715, 16sylibr 133 1  |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   U_ciun 3871   Tr wtr 4085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-iun 3873  df-tr 4086
This theorem is referenced by:  truni  4099
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