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Theorem triun 4307
 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 4089 . . . 4
2 r19.29 2838 . . . . 5
3 nfcv 2571 . . . . . . 7
4 nfiu1 4113 . . . . . . 7
53, 4nfss 3333 . . . . . 6
6 trss 4303 . . . . . . . 8
76imp 419 . . . . . . 7
8 ssiun2 4126 . . . . . . . 8
9 sstr2 3347 . . . . . . . 8
108, 9syl5com 28 . . . . . . 7
117, 10syl5 30 . . . . . 6
125, 11rexlimi 2815 . . . . 5
132, 12syl 16 . . . 4
141, 13sylan2b 462 . . 3
1514ralrimiva 2781 . 2
16 dftr3 4298 . 2
1715, 16sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  wral 2697  wrex 2698   wss 3312  ciun 4085   wtr 4294 This theorem is referenced by:  truni  4308  r1tr  7694  r1elssi  7723 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-iun 4087  df-tr 4295
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