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Theorem triun 3971
 Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
triun
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem triun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eliun 3756 . . . 4
2 r19.29 2520 . . . . 5
3 nfcv 2235 . . . . . . 7
4 nfiu1 3782 . . . . . . 7
53, 4nfss 3032 . . . . . 6
6 trss 3967 . . . . . . . 8
76imp 123 . . . . . . 7
8 ssiun2 3795 . . . . . . . 8
9 sstr2 3046 . . . . . . . 8
108, 9syl5com 29 . . . . . . 7
117, 10syl5 32 . . . . . 6
125, 11rexlimi 2495 . . . . 5
132, 12syl 14 . . . 4
141, 13sylan2b 282 . . 3
1514ralrimiva 2458 . 2
16 dftr3 3962 . 2
1715, 16sylibr 133 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wcel 1445  wral 2370  wrex 2371   wss 3013  ciun 3752   wtr 3958 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-uni 3676  df-iun 3754  df-tr 3959 This theorem is referenced by:  truni  3972
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