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Theorem triin 4958
Description: An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
Assertion
Ref Expression
triin (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Proof of Theorem triin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3386 . . . . 5 𝑦 ∈ V
2 eliin 4713 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 r19.26 3243 . . . . . . 7 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) ↔ (∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵))
54biimpri 220 . . . . . 6 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 (Tr 𝐵𝑦𝐵))
6 trss 4952 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
76imp 396 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
87ralimi 3131 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
95, 8syl 17 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
10 ssiin 4758 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
119, 10sylibr 226 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
123, 11sylan2b 588 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1312ralrimiva 3145 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
14 dftr3 4947 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1513, 14sylibr 226 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wral 3087  Vcvv 3383  wss 3767   ciin 4709  Tr wtr 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-v 3385  df-in 3774  df-ss 3781  df-uni 4627  df-iin 4711  df-tr 4944
This theorem is referenced by:  trint  4959
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