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Theorem triun 3858
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
triun  Tr  Tr  U_
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem triun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eliun 3652 . . . 4  U_
2 r19.29 2444 . . . . 5  Tr  Tr
3 nfcv 2175 . . . . . . 7  F/_
4 nfiu1 3678 . . . . . . 7  F/_ U_
53, 4nfss 2932 . . . . . 6  F/  C_  U_
6 trss 3854 . . . . . . . 8  Tr  C_
76imp 115 . . . . . . 7  Tr  C_
8 ssiun2 3691 . . . . . . . 8  C_ 
U_
9 sstr2 2946 . . . . . . . 8 
C_  C_  U_  C_ 
U_
108, 9syl5com 26 . . . . . . 7  C_  C_ 
U_
117, 10syl5 28 . . . . . 6  Tr  C_ 
U_
125, 11rexlimi 2420 . . . . 5  Tr  C_ 
U_
132, 12syl 14 . . . 4  Tr  C_  U_
141, 13sylan2b 271 . . 3  Tr 
U_  C_ 
U_
1514ralrimiva 2386 . 2  Tr 
U_  C_  U_
16 dftr3 3849 . 2  Tr 
U_  U_  C_  U_
1715, 16sylibr 137 1  Tr  Tr  U_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1390  wral 2300  wrex 2301    C_ wss 2911   U_ciun 3648   Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-iun 3650  df-tr 3846
This theorem is referenced by:  truni  3859
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