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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 (x A Tr B → Tr x A B)
Distinct variable group:   x,A
Allowed substitution hint:   B(x)

Proof of Theorem triun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eliun 3652 . . . 4 (y x A Bx A y B)
2 r19.29 2444 . . . . 5 ((x A Tr B x A y B) → x A (Tr B y B))
3 nfcv 2175 . . . . . . 7 xy
4 nfiu1 3678 . . . . . . 7 x x A B
53, 4nfss 2932 . . . . . 6 x y x A B
6 trss 3854 . . . . . . . 8 (Tr B → (y ByB))
76imp 115 . . . . . . 7 ((Tr B y B) → yB)
8 ssiun2 3691 . . . . . . . 8 (x AB x A B)
9 sstr2 2946 . . . . . . . 8 (yB → (B x A By x A B))
108, 9syl5com 26 . . . . . . 7 (x A → (yBy x A B))
117, 10syl5 28 . . . . . 6 (x A → ((Tr B y B) → y x A B))
125, 11rexlimi 2420 . . . . 5 (x A (Tr B y B) → y x A B)
132, 12syl 14 . . . 4 ((x A Tr B x A y B) → y x A B)
141, 13sylan2b 271 . . 3 ((x A Tr B y x A B) → y x A B)
1514ralrimiva 2386 . 2 (x A Tr By x A By x A B)
16 dftr3 3849 . 2 (Tr x A By x A By x A B)
1715, 16sylibr 137 1 (x A Tr B → Tr x A B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ∪ 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-bnd 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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