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Theorem triun 5274
Description: An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
triun (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Proof of Theorem triun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4995 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.29 3114 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 (Tr 𝐵𝑦𝐵))
3 nfcv 2905 . . . . . . 7 𝑥𝑦
4 nfiu1 5027 . . . . . . 7 𝑥 𝑥𝐴 𝐵
53, 4nfss 3976 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
6 trss 5270 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 406 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
8 ssiun2 5047 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
9 sstr2 3990 . . . . . . 7 (𝑦𝐵 → (𝐵 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
107, 8, 9syl2imc 41 . . . . . 6 (𝑥𝐴 → ((Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵))
115, 10rexlimi 3259 . . . . 5 (∃𝑥𝐴 (Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
122, 11syl 17 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
131, 12sylan2b 594 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1413ralrimiva 3146 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
15 dftr3 5265 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1614, 15sylibr 234 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  wrex 3070  wss 3951   ciun 4991  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-iun 4993  df-tr 5260
This theorem is referenced by:  truni  5275  r1tr  9816  r1elssi  9845  iunord  49195
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