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

Theorem trint 5191
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.) (Proof shortened by BJ, 3-Oct-2022.)
Assertion
Ref Expression
trint (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trint
StepHypRef Expression
1 triin 5190 . 2 (∀𝑥𝐴 Tr 𝑥 → Tr 𝑥𝐴 𝑥)
2 intiin 4982 . . 3 𝐴 = 𝑥𝐴 𝑥
3 treq 5181 . . 3 ( 𝐴 = 𝑥𝐴 𝑥 → (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥))
42, 3ax-mp 5 . 2 (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥)
51, 4sylibr 237 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wral 3062   cint 4873   ciin 4919  Tr wtr 5175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-v 3422  df-in 3887  df-ss 3897  df-uni 4834  df-int 4874  df-iin 4921  df-tr 5176
This theorem is referenced by:  tctr  9380  intwun  10373  intgru  10452  dfon2lem8  33508
  Copyright terms: Public domain W3C validator