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Theorem triun 4558
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:   ,

Proof of Theorem triun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eliun 4335 . . . 4
2 r19.29 2992 . . . . 5
3 nfcv 2619 . . . . . . 7
4 nfiu1 4360 . . . . . . 7
53, 4nfss 3496 . . . . . 6
6 trss 4554 . . . . . . . 8
76imp 429 . . . . . . 7
8 ssiun2 4373 . . . . . . . 8
9 sstr2 3510 . . . . . . . 8
108, 9syl5com 30 . . . . . . 7
117, 10syl5 32 . . . . . 6
125, 11rexlimi 2939 . . . . 5
132, 12syl 16 . . . 4
141, 13sylan2b 475 . . 3
1514ralrimiva 2871 . 2
16 dftr3 4549 . 2
1715, 16sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  U_ciun 4330  Trwtr 4545
This theorem is referenced by:  truni  4559  r1tr  8215  r1elssi  8244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-iun 4332  df-tr 4546
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