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Theorem triun 5241
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 4962 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.29 3114 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 (Tr 𝐵𝑦𝐵))
3 nfcv 2904 . . . . . . 7 𝑥𝑦
4 nfiu1 4992 . . . . . . 7 𝑥 𝑥𝐴 𝐵
53, 4nfss 3940 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
6 trss 5237 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 408 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
8 ssiun2 5011 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
9 sstr2 3955 . . . . . . 7 (𝑦𝐵 → (𝐵 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
107, 8, 9syl2imc 41 . . . . . 6 (𝑥𝐴 → ((Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵))
115, 10rexlimi 3241 . . . . 5 (∃𝑥𝐴 (Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
122, 11syl 17 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
131, 12sylan2b 595 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1413ralrimiva 3140 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
15 dftr3 5232 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1614, 15sylibr 233 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  wrex 3070  wss 3914   ciun 4958  Tr wtr 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-v 3449  df-in 3921  df-ss 3931  df-uni 4870  df-iun 4960  df-tr 5227
This theorem is referenced by:  truni  5242  r1tr  9720  r1elssi  9749  iunord  47211
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