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Theorem trin 4915
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 3939 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2 trss 4913 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
3 trss 4913 . . . . . 6 (Tr 𝐵 → (𝑥𝐵𝑥𝐵))
42, 3im2anan9 916 . . . . 5 ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵)))
51, 4syl5bi 232 . . . 4 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → (𝑥𝐴𝑥𝐵)))
6 ssin 3978 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
75, 6syl6ib 241 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴𝐵) → 𝑥 ⊆ (𝐴𝐵)))
87ralrimiv 3103 . 2 ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
9 dftr3 4908 . 2 (Tr (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥 ⊆ (𝐴𝐵))
108, 9sylibr 224 1 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  cin 3714  wss 3715  Tr wtr 4904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905
This theorem is referenced by:  ordin  5914  tcmin  8792  ingru  9849  gruina  9852  dfon2lem4  32017
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