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Theorem triun 4503
 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 4274 . . . 4
2 r19.29 2912 . . . . 5
3 nfcv 2612 . . . . . . 7
4 nfiu1 4299 . . . . . . 7
53, 4nfss 3411 . . . . . 6
6 trss 4499 . . . . . . . 8
76imp 436 . . . . . . 7
8 ssiun2 4312 . . . . . . . 8
9 sstr2 3425 . . . . . . . 8
108, 9syl5com 30 . . . . . . 7
117, 10syl5 32 . . . . . 6
125, 11rexlimi 2864 . . . . 5
132, 12syl 17 . . . 4
141, 13sylan2b 483 . . 3
1514ralrimiva 2809 . 2
16 dftr3 4494 . 2
1715, 16sylibr 217 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wcel 1904  wral 2756  wrex 2757   wss 3390  ciun 4269   wtr 4490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-iun 4271  df-tr 4491 This theorem is referenced by:  truni  4504  r1tr  8265  r1elssi  8294
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