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Theorem trin 5233
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 3925 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 trss 5232 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
3 trss 5232 . . . . . 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 4189 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
75, 6syl6ib 251 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → 𝑥 ⊆ (𝐴𝐵)))
87ralrimiv 3141 . 2 ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
9 dftr3 5227 . 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 3063  cin 3908  wss 3909  Tr wtr 5221
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 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-v 3446  df-in 3916  df-ss 3926  df-uni 4865  df-tr 5222
This theorem is referenced by:  ordin  6346  tcmin  9636  ingru  10710  gruina  10713  dfon2lem4  34171
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