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Theorem trint 4696
 Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
Assertion
Ref Expression
trint (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trint
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dftr3 4684 . . . . 5 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
21ralbii 2963 . . . 4 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
3 df-ral 2901 . . . . . 6 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦𝑥))
43ralbii 2963 . . . . 5 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥))
5 ralcom4 3197 . . . . 5 (∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
64, 5bitri 263 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
72, 6sylbb 208 . . 3 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
8 ralim 2932 . . 3 (∀𝑥𝐴 (𝑦𝑥𝑦𝑥) → (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
97, 8sylg 1740 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
10 dftr3 4684 . . 3 (Tr 𝐴 ↔ ∀𝑦 𝐴𝑦 𝐴)
11 df-ral 2901 . . . 4 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(𝑦 𝐴𝑦 𝐴))
12 vex 3176 . . . . . . 7 𝑦 ∈ V
1312elint2 4417 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
14 ssint 4428 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
1513, 14imbi12i 339 . . . . 5 ((𝑦 𝐴𝑦 𝐴) ↔ (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1615albii 1737 . . . 4 (∀𝑦(𝑦 𝐴𝑦 𝐴) ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1711, 16bitri 263 . . 3 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1810, 17bitri 263 . 2 (Tr 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
199, 18sylibr 223 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∩ cint 4410  Tr wtr 4680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-int 4411  df-tr 4681 This theorem is referenced by:  tctr  8499  intwun  9436  intgru  9515  dfon2lem8  30939
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