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Theorem trint 4733
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 4721 . . . . 5 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
21ralbii 2975 . . . 4 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
3 df-ral 2912 . . . . . 6 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦𝑥))
43ralbii 2975 . . . . 5 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥))
5 ralcom4 3213 . . . . 5 (∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
64, 5bitri 264 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
72, 6sylbb 209 . . 3 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
8 ralim 2943 . . 3 (∀𝑥𝐴 (𝑦𝑥𝑦𝑥) → (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
97, 8sylg 1747 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
10 dftr3 4721 . . 3 (Tr 𝐴 ↔ ∀𝑦 𝐴𝑦 𝐴)
11 df-ral 2912 . . . 4 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(𝑦 𝐴𝑦 𝐴))
12 vex 3192 . . . . . . 7 𝑦 ∈ V
1312elint2 4452 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
14 ssint 4463 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
1513, 14imbi12i 340 . . . . 5 ((𝑦 𝐴𝑦 𝐴) ↔ (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1615albii 1744 . . . 4 (∀𝑦(𝑦 𝐴𝑦 𝐴) ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1711, 16bitri 264 . . 3 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1810, 17bitri 264 . 2 (Tr 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
199, 18sylibr 224 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wcel 1987  wral 2907  wss 3559   cint 4445  Tr wtr 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3191  df-in 3566  df-ss 3573  df-uni 4408  df-int 4446  df-tr 4718
This theorem is referenced by:  tctr  8567  intwun  9508  intgru  9587  dfon2lem8  31423
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