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

Theorem trin 5238
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
trin ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))

Proof of Theorem trin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3930 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 trss 5237 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
3 trss 5237 . . . . . 6 (Tr 𝐵 → (𝑥𝐵𝑥𝐵))
42, 3im2anan9 621 . . . . 5 ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵)))
51, 4biimtrid 241 . . . 4 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → (𝑥𝐴𝑥𝐵)))
6 ssin 4194 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
75, 6syl6ib 251 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → 𝑥 ⊆ (𝐴𝐵)))
87ralrimiv 3139 . 2 ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
9 dftr3 5232 . 2 (Tr (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
108, 9sylibr 233 1 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  cin 3913  wss 3914  Tr wtr 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-uni 4870  df-tr 5227
This theorem is referenced by:  ordin  6351  tcmin  9685  ingru  10759  gruina  10762  dfon2lem4  34424
  Copyright terms: Public domain W3C validator