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Theorem trin 5271
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 3967 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 trss 5270 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
3 trss 5270 . . . . . 6 (Tr 𝐵 → (𝑥𝐵𝑥𝐵))
42, 3im2anan9 620 . . . . 5 ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵)))
51, 4biimtrid 242 . . . 4 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → (𝑥𝐴𝑥𝐵)))
6 ssin 4239 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
75, 6imbitrdi 251 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → 𝑥 ⊆ (𝐴𝐵)))
87ralrimiv 3145 . 2 ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
9 dftr3 5265 . 2 (Tr (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
108, 9sylibr 234 1 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  cin 3950  wss 3951  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-tr 5260
This theorem is referenced by:  ordin  6414  tcmin  9781  ingru  10855  gruina  10858  dfon2lem4  35787
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