Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  triun Structured version   Visualization version   GIF version

Theorem triun 5150
 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 4886 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
2 r19.29 3216 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → ∃𝑥𝐴 (Tr 𝐵𝑦𝐵))
3 nfcv 2955 . . . . . . 7 𝑥𝑦
4 nfiu1 4916 . . . . . . 7 𝑥 𝑥𝐴 𝐵
53, 4nfss 3907 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
6 trss 5146 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 410 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
8 ssiun2 4935 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
9 sstr2 3922 . . . . . . 7 (𝑦𝐵 → (𝐵 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵))
107, 8, 9syl2imc 41 . . . . . 6 (𝑥𝐴 → ((Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵))
115, 10rexlimi 3274 . . . . 5 (∃𝑥𝐴 (Tr 𝐵𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
122, 11syl 17 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∃𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
131, 12sylan2b 596 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1413ralrimiva 3149 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
15 dftr3 5141 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1614, 15sylibr 237 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ ciun 4882  Tr wtr 5137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802  df-iun 4884  df-tr 5138 This theorem is referenced by:  truni  5151  r1tr  9192  r1elssi  9221  iunord  45247
 Copyright terms: Public domain W3C validator