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Theorem trin 3905
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 3165 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 trss 3904 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
3 trss 3904 . . . . . 6 (Tr 𝐵 → (𝑥𝐵𝑥𝐵))
42, 3im2anan9 563 . . . . 5 ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵)))
51, 4syl5bi 150 . . . 4 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → (𝑥𝐴𝑥𝐵)))
6 ssin 3204 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
75, 6syl6ib 159 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → 𝑥 ⊆ (𝐴𝐵)))
87ralrimiv 2438 . 2 ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
9 dftr3 3899 . 2 (Tr (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
108, 9sylibr 132 1 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  wral 2353  cin 2981  wss 2982  Tr wtr 3895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896
This theorem is referenced by:  ordin  4168
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