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Theorem trint 5174
 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 5173 . 2 (∀𝑥𝐴 Tr 𝑥 → Tr 𝑥𝐴 𝑥)
2 intiin 4969 . . 3 𝐴 = 𝑥𝐴 𝑥
3 treq 5164 . . 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 1538  ∀wral 3133  ∩ cint 4862  ∩ ciin 4906  Tr wtr 5158 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-in 3926  df-ss 3936  df-uni 4825  df-int 4863  df-iin 4908  df-tr 5159 This theorem is referenced by:  tctr  9179  intwun  10155  intgru  10234  dfon2lem8  33095
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